Difference between revisions of "Groups of order 16"
(→Element structure) |
(→Permutation representations: added permutation representations for remaining groups) |
||
Line 118: | Line 118: | ||
| [[direct product of Z4 and Z4]] || 2 || 3 || <math>\langle (1,2,3,4), (5,6,7,8) \rangle</math> | | [[direct product of Z4 and Z4]] || 2 || 3 || <math>\langle (1,2,3,4), (5,6,7,8) \rangle</math> | ||
|- | |- | ||
− | | [[SmallGroup(16,3)]] || 3 || 9 || | + | | [[SmallGroup(16,3)]] || 3 || 9 || <math>\langle (1,2,3,4),(1,3)(5,6,7,8) \rangle</math> |
|- | |- | ||
| [[nontrivial semidirect product of Z4 and Z4]] || 4 || 10 || <math>\langle (1,2,3,4), (1,3)(5,6,7,8)\rangle</math> | | [[nontrivial semidirect product of Z4 and Z4]] || 4 || 10 || <math>\langle (1,2,3,4), (1,3)(5,6,7,8)\rangle</math> | ||
Line 130: | Line 130: | ||
| [[semidihedral group:SD16]] || 8 || 13 || <math>\langle (1,2,3,4,5,6,7,8), (1,3)(5,7)(2,6) \rangle</math> | | [[semidihedral group:SD16]] || 8 || 13 || <math>\langle (1,2,3,4,5,6,7,8), (1,3)(5,7)(2,6) \rangle</math> | ||
|- | |- | ||
− | | [[generalized quaternion group:Q16]] || 9 || 14 || | + | | [[generalized quaternion group:Q16]] || 9 || 14 || <math>\langle (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,9,5,13)(2,16,6,12)(3,15,7,11)(4,14,8,10) \rangle</math> |
|- | |- | ||
| [[direct product of Z4 and V4]] || 10 || 2 || <math>\langle (1,2,3,4), (5,6), (7,8)\rangle</math> | | [[direct product of Z4 and V4]] || 10 || 2 || <math>\langle (1,2,3,4), (5,6), (7,8)\rangle</math> | ||
Line 136: | Line 136: | ||
| [[direct product of D8 and Z2]] || 11 || 6 || <math>\langle (1,2,3,4), (1,3), (5,6)\rangle</math> | | [[direct product of D8 and Z2]] || 11 || 6 || <math>\langle (1,2,3,4), (1,3), (5,6)\rangle</math> | ||
|- | |- | ||
− | | [[direct product of Q8 and Z2]] || 12 || 7 || | + | | [[direct product of Q8 and Z2]] || 12 || 7 || <math>\langle (1,2,4,7)(3,6,8,5),(1,3,4,8)(2,5,7,6),(9,10) \rangle</math> |
|- | |- | ||
− | | [[central product of D8 and Z4]] || 13 || 8 || | + | | [[central product of D8 and Z4]] || 13 || 8 || <math>\langle (1,2,3,4)(5,6,7,8),(1,7)(2,8)(3,5)(4,6),(1,6,3,8)(2,5,4,7) \rangle</math> |
|- | |- | ||
| [[elementary abelian group:E16]] || 14 || 1 || <math>\langle (1,2), (3,4), (5,6), (7,8)\rangle</math> | | [[elementary abelian group:E16]] || 14 || 1 || <math>\langle (1,2), (3,4), (5,6), (7,8)\rangle</math> | ||
Line 149: | Line 149: | ||
gap> IdGroup(Group([(1,2,3,4),(5,6,7,8)])); | gap> IdGroup(Group([(1,2,3,4),(5,6,7,8)])); | ||
[ 16, 2 ] | [ 16, 2 ] | ||
+ | gap> IdGroup(Group([(1,2)(3,4),(1,3)(2,4),(1,3)(5,6,7,8)])); | ||
+ | [ 16, 3 ] | ||
gap> IdGroup(Group([(1,2,3,4),(1,3)(5,6,7,8)])); | gap> IdGroup(Group([(1,2,3,4),(1,3)(5,6,7,8)])); | ||
[ 16, 4 ] | [ 16, 4 ] | ||
Line 159: | Line 161: | ||
gap> IdGroup(Group([(1,2,3,4,5,6,7,8),(1,3)(5,7)(2,6)])); | gap> IdGroup(Group([(1,2,3,4,5,6,7,8),(1,3)(5,7)(2,6)])); | ||
[ 16, 8 ] | [ 16, 8 ] | ||
+ | gap> IdGroup(Group([(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16),(1,9,5,13)(2,16,6,12)(3,15,7,11)(4,14,8,10)])); | ||
+ | [ 16, 9 ] | ||
gap> IdGroup(Group([(1,2,3,4),(5,6),(7,8)])); | gap> IdGroup(Group([(1,2,3,4),(5,6),(7,8)])); | ||
[ 16, 10 ] | [ 16, 10 ] | ||
gap> IdGroup(Group([(1,2,3,4),(1,3),(5,6)])); | gap> IdGroup(Group([(1,2,3,4),(1,3),(5,6)])); | ||
[ 16, 11 ] | [ 16, 11 ] | ||
+ | gap> IdGroup(Group([(1,2,4,7)(3,6,8,5),(1,3,4,8)(2,5,7,6),(9,10)])); | ||
+ | [ 16, 12 ] | ||
+ | gap> IdGroup(Group([(1,2,3,4)(5,6,7,8),(1,7)(2,8)(3,5)(4,6),(1,6,3,8)(2,5,4,7)])); | ||
+ | [ 16, 13 ] | ||
gap> IdGroup(Group([(1,2),(3,4),(5,6),(7,8)])); | gap> IdGroup(Group([(1,2),(3,4),(5,6),(7,8)])); | ||
[ 16, 14 ]</pre></toggledisplay> | [ 16, 14 ]</pre></toggledisplay> |
Revision as of 23:51, 15 April 2017
Contents |
This article gives information about, and links to more details on, groups of order 16
See pages on algebraic structures of order 16| See pages on groups of a particular order
This article gives basic information comparing and contrasting groups of order 16. See also more detailed information on specific subtopics through the links:
Information type | Page summarizing information for groups of order 16 |
---|---|
element structure (element orders, conjugacy classes, etc.) | element structure of groups of order 16 |
subgroup structure | subgroup structure of groups of order 16 |
linear representation theory | linear representation theory of groups of order 16 projective representation theory of groups of order 16 modular representation theory of groups of order 16 |
endomorphism structure, automorphism structure | endomorphism structure of groups of order 16 |
group cohomology | group cohomology of groups of order 16 |
Statistics at a glance
To understand these in a broader context, see
groups of order 2^n |groups of prime-fourth order
Numbers of groups
Quantity | Value | Explanation |
---|---|---|
Total number of groups | 14 | |
Number of abelian groups | 5 | equals the number of unordered integer partitions of 4. See classification of finite abelian groups and structure theorem for finitely generated abelian groups. |
Number of non-abelian groups of class two | 6 | |
Number of groups of nilpotency class exactly three, i.e., maximal class groups | 3 | classification of finite 2-groups of maximal class. For order , there are exactly three maximal class groups: dihedral, semidihedral, and generalized quaternion. For order 16, the groups are: dihedral group:D16, semidihedral group:SD16, and generalized quaternion group:Q16. |
Numbers of equivalence classes of groups
Equivalence relation on groups | Number of equivalence classes of groups of order 16 | Sizes of equivalence classes, i.e., number of isomorphism classes of groups within each equivalence class (should add up to 14) | More information |
---|---|---|---|
isoclinic groups (i.e., Hall-Senior families) | 3 | 5,6,3 | #Families and classification, see also classification of groups of order 16. For order 16, the nilpotency class is sufficient to determine the group up to isoclinism. |
having the same conjugacy class size statistics | 3 | 5,6,3 | Element structure of groups of order 16#Conjugacy class sizes. For order 16, as for other small prime power orders, the nilpotency class determines the conjugacy class size statistics, and vice versa. See nilpotency class and order determine conjugacy class size statistics for groups up to prime-fourth order. |
having the same degrees of irreducible representations | 3 | 5,6,3 | See Linear representation theory of groups of order 16#Degrees of irreducible representations. Note that for order 16, as for other small prime power orders, the nilpotency class determines the conjugacy class size statistics, and vice versa. See nilpotency class and order determine conjugacy class size statistics for groups up to prime-fourth order. |
1-isomorphic groups | 12 | 1 (10 times), 2 (2 times) | Element structure of groups of order 16#1-isomorphism |
The list
IDs and symbols
Group | Second part of GAP ID (GAP ID is (16,second part)) | Hall-Senior number (among groups of order 16) | Nilpotency class | Hall-Senior symbol | Probability in cohomology tree probability distribution (as proper fraction) | Probability in cohomology tree probability distribution (as numerical value) |
---|---|---|---|---|---|---|
cyclic group:Z16 | 1 | 5 | 1 | 1/8 | 0.1250 | |
direct product of Z4 and Z4 | 2 | 3 | 1 | 7/128 | 0.0547 | |
SmallGroup(16,3) | 3 | 9 | 2 | 13/128 | 0.1016 | |
nontrivial semidirect product of Z4 and Z4 | 4 | 10 | 2 | 1/8 | 0.1250 | |
direct product of Z8 and Z2 | 5 | 4 | 1 | 15/64 | 0.2344 | |
M16 | 6 | 11 | 2 | 7/64 | 0.1094 | |
dihedral group:D16 | 7 | 12 | 3 | 3/128 | 0.0234 | |
semidihedral group:SD16 | 8 | 13 | 3 | 3/64 | 0.0469 | |
generalized quaternion group:Q16 | 9 | 14 | 3 | 3/128 | 0.0234 | |
direct product of Z4 and V4 | 10 | 2 | 1 | 63/1024 | 0.0615 | |
direct product of D8 and Z2 | 11 | 6 | 2 | 45/1024 | 0.0440 | |
direct product of Q8 and Z2 | 12 | 7 | 2 | 23/1024 | 0.0225 | |
central product of D8 and Z4 | 13 | 8 | 2 | 7/256 | 0.0273 | |
elementary abelian group:E16 | 14 | 1 | 1 | 1/1024 | 0.0010 |
Presentations
In these presentations, we use for the identity element.
Group | Second part of GAP ID (GAP ID is (16,second part)) | Hall-Senior number (among groups of order 16) | Presentation |
---|---|---|---|
cyclic group:Z16 | 1 | 5 | |
direct product of Z4 and Z4 | 2 | 3 | |
SmallGroup(16,3) | 3 | 9 | |
nontrivial semidirect product of Z4 and Z4 | 4 | 10 | |
direct product of Z8 and Z2 | 5 | 4 | |
M16 | 6 | 11 | |
dihedral group:D16 | 7 | 12 | |
semidihedral group:SD16 | 8 | 13 | |
generalized quaternion group:Q16 | 9 | 14 | |
direct product of Z4 and V4 | 10 | 2 | |
direct product of D8 and Z2 | 11 | 6 | |
direct product of Q8 and Z2 | 12 | 7 | (check) |
central product of D8 and Z4 | 13 | 8 | |
elementary abelian group:E16 | 14 | 1 |
Permutation representations
These give faithful permutation representations of the group:
Group | Second part of GAP ID (GAP ID is (16,second part)) | Hall-Senior number (among groups of order 16) | Permutation representation |
---|---|---|---|
cyclic group:Z16 | 1 | 5 | |
direct product of Z4 and Z4 | 2 | 3 | |
SmallGroup(16,3) | 3 | 9 | |
nontrivial semidirect product of Z4 and Z4 | 4 | 10 | |
direct product of Z8 and Z2 | 5 | 4 | |
M16 | 6 | 11 | |
dihedral group:D16 | 7 | 12 | |
semidihedral group:SD16 | 8 | 13 | |
generalized quaternion group:Q16 | 9 | 14 | |
direct product of Z4 and V4 | 10 | 2 | |
direct product of D8 and Z2 | 11 | 6 | |
direct product of Q8 and Z2 | 12 | 7 | |
central product of D8 and Z4 | 13 | 8 | |
elementary abelian group:E16 | 14 | 1 |
Subgroup/quotient relationships
Subgroup relationships
Quotient relationships
Arithmetic functions
Summary information
Here, the rows are arithmetic functions that take values between and , and the columns give the possible values of these functions. The entry in each cell is the number of isomorphism classes of groups for which the row arithmetic function takes the column value. Note that all the row value sums must equal .
Arithmetic function | Value 0 | Value 1 | Value 2 | Value 3 | Value 4 |
---|---|---|---|---|---|
prime-base logarithm of exponent | 0 | 1 | 7 | 5 | 1 |
nilpotency class | 0 | 5 | 6 | 3 | 0 |
derived length | 0 | 5 | 9 | 0 | 0 |
Frattini length | 0 | 1 | 7 | 5 | 1 |
minimum size of generating set | 0 | 1 | 8 | 4 | 1 |
number of factors in direct product decomposition | 0 | 8 | 4 | 1 | 1 |
Functions taking values between 0 and 4
These measure ranks of subgroups, lengths of series, or the prime-base logarithms of orders of certain subgroups.
Group | GAP ID second part | Hall-Senior number | prime-base logarithm of exponent | nilpotency class | derived length | Frattini length | minimum size of generating set | subgroup rank | rank | normal rank | characteristic rank | prime-base logarithm of order of derived subgroup | prime-base logarithm of order of inner automorphism group | number of factors in complete direct product decomposition |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
cyclic group:Z16 | 1 | 5 | 4 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 |
direct product of Z4 and Z4 | 2 | 3 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 |
SmallGroup(16,3) | 3 | 9 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 1 | 2 | 1 |
nontrivial semidirect product of Z4 and Z4 | 4 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 1 |
direct product of Z8 and Z2 | 5 | 4 | 3 | 1 | 1 | 3 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 |
M16 | 6 | 11 | 3 | 2 | 2 | 3 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 1 |
dihedral group:D16 | 7 | 12 | 3 | 3 | 2 | 3 | 2 | 2 | 2 | 1 | 1 | 2 | 3 | 1 |
semidihedral group:SD16 | 8 | 13 | 3 | 3 | 2 | 3 | 2 | 2 | 2 | 1 | 1 | 2 | 3 | 1 |
generalized quaternion group:Q16 | 9 | 14 | 3 | 3 | 2 | 3 | 2 | 2 | 1 | 1 | 1 | 2 | 3 | 1 |
direct product of Z4 and V4 | 10 | 2 | 2 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 0 | 0 | 3 |
direct product of D8 and Z2 | 11 | 6 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 2 | 1 | 2 | 2 |
direct product of Q8 and Z2 | 12 | 7 | 2 | 2 | 2 | 2 | 3 | 3 | 2 | 2 | 2 | 1 | 2 | 2 |
central product of D8 and Z4 | 13 | 8 | 2 | 2 | 2 | 2 | 3 | 3 | 2 | 2 | 1 | 1 | 2 | 1 |
elementary abelian group:E16 | 14 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 0 | 0 | 4 |
Same, with rows and columns interchanged:
Function | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
prime-base logarithm of exponent | 4 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | 1 |
nilpotency class | 1 | 1 | 2 | 2 | 1 | 2 | 3 | 3 | 3 | 1 | 2 | 2 | 2 | 1 |
derived length | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 1 |
Frattini length | 4 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | 1 |
minimum size of generating set | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 |
subgroup rank | 1 | 2 | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 |
rank as p-group | 1 | 2 | 3 | 2 | 2 | 2 | 2 | 2 | 1 | 3 | 3 | 2 | 2 | 4 |
normal rank as p-group | 1 | 2 | 3 | 2 | 2 | 2 | 1 | 1 | 1 | 3 | 3 | 2 | 2 | 4 |
characteristic rank as p-group | 1 | 2 | 3 | 2 | 2 | 2 | 1 | 1 | 1 | 3 | 2 | 2 | 1 | 4 |
prime-base logarithm of order of derived subgroup | 0 | 0 | 1 | 1 | 0 | 1 | 2 | 2 | 2 | 0 | 1 | 1 | 1 | 0 |
prime-base logarithm of order of inner automorphism group | 0 | 0 | 2 | 2 | 0 | 2 | 3 | 3 | 3 | 0 | 2 | 2 | 2 | 0 |
Arithmetic function values of a counting nature
Group | GAP ID second part | Hall-Senior number | number of conjugacy classes | number of subgroups | number of conjugacy classes of subgroups | number of normal subgroups | number of characteristic subgroups | number of automorphism classes of subgroups |
---|---|---|---|---|---|---|---|---|
cyclic group:Z16 | 1 | 5 | 16 | 5 | 5 | 5 | 5 | 5 |
direct product of Z4 and Z4 | 2 | 3 | 16 | 15 | 15 | 15 | 3 | |
SmallGroup(16,3) | 3 | 9 | 10 | 23 | 17 | 11 | 5 | |
nontrivial semidirect product of Z4 and Z4 | 4 | 10 | 10 | 15 | 13 | 11 | 7 | |
direct product of Z8 and Z2 | 5 | 4 | 16 | 11 | 11 | 11 | 7 | |
M16 | 6 | 11 | 10 | 11 | 10 | 9 | 7 | |
dihedral group:D16 | 7 | 12 | 7 | 19 | 11 | 7 | 5 | 9 |
semidihedral group:SD16 | 8 | 13 | 7 | 15 | 10 | 7 | 7 | |
quaternion group | 9 | 14 | 7 | 11 | 9 | 7 | 5 | |
direct product of Z4 and V4 | 10 | 2 | 16 | 27 | 27 | 27 | 4 | |
direct product of D8 and Z2 | 11 | 6 | 10 | 35 | 27 | 19 | 5 | |
direct product of Q8 and Z2 | 12 | 7 | 10 | 19 | 19 | 19 | 4 | |
central product of D8 and Z4 | 13 | 8 | 10 | 23 | 20 | 17 | 5 | |
elementary abelian group:E16 | 14 | 1 | 16 | 67 | 67 | 67 | 2 | 5 |
Same, with rows and columns interchanged:
Function | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
number of conjugacy classes | 16 | 16 | 10 | 10 | 16 | 10 | 7 | 7 | 7 | 16 | 10 | 10 | 10 | 16 |
number of subgroups | 5 | 15 | 23 | 15 | 11 | 11 | 19 | 15 | 11 | 27 | 35 | 19 | 23 | 67 |
number of conjugacy classes of subgroups | 5 | 15 | 17 | 13 | 11 | 10 | 11 | 10 | 9 | 27 | 27 | 19 | 20 | 67 |
number of characteristic subgroups | 5 | 3 | 5 | 7 | 7 | 7 | 5 | 7 | 5 | 4 | 5 | 4 | 5 | 2 |
Numerical invariants
Here first is the table with the number of conjugacy classes of each size (for more detailed breakdown information, see element structure of groups of order 16). Note also that nilpotency class and order determine conjugacy class size statistics for groups up to prime-fourth order, so that there are only three types -- the abelian groups, the non-abelian groups of class two, and the groups of class exactly three:
Group | GAP ID | Nilpotency class | Conjugacy classes of size 1 | Conjugacy classes of size 2 | Conjugacy classes of size 4 |
---|---|---|---|---|---|
cyclic group:Z16 | 1 | 1 | 16 | 0 | 0 |
direct product of Z4 and Z4 | 2 | 1 | 16 | 0 | 0 |
SmallGroup(16,3) | 3 | 2 | 4 | 6 | 0 |
nontrivial semidirect product of Z4 and Z4 | 4 | 2 | 4 | 6 | 0 |
direct product of Z8 and Z2 | 5 | 1 | 16 | 0 | 0 |
M16 | 6 | 2 | 4 | 6 | 0 |
dihedral group:D16 | 7 | 3 | 2 | 3 | 2 |
semidihedral group:SD16 | 8 | 3 | 2 | 3 | 2 |
generalized quaternion group:Q16 | 9 | 3 | 2 | 3 | 2 |
direct product of Z4 and V4 | 10 | 1 | 16 | 0 | 0 |
direct product of D8 and Z2 | 11 | 2 | 4 | 6 | 0 |
direct product of Q8 and Z2 | 12 | 2 | 4 | 6 | 0 |
central product of D8 and Z4 | 13 | 2 | 4 | 6 | 0 |
elementary abelian group:E16 | 14 | 1 | 16 | 0 | 0 |
Here are the degrees of irreducible representations. Note also that nilpotency class and order determine degrees of irreducible representations for groups up to prime-fourth order, so that there are only three types -- the abelian groups, the non-abelian groups of class two, and the groups of class exactly three:
Group | GAP ID | Nilpotency class | Degree 1 | Degree 2 | Degree 4 |
---|---|---|---|---|---|
cyclic group:Z16 | 1 | 1 | 16 | 0 | 0 |
direct product of Z4 and Z4 | 2 | 1 | 16 | 0 | 0 |
SmallGroup(16,3) | 3 | 2 | 8 | 2 | 0 |
nontrivial semidirect product of Z4 and Z4 | 4 | 2 | 8 | 2 | 0 |
direct product of Z8 and Z2 | 5 | 1 | 16 | 0 | 0 |
M16 | 6 | 2 | 8 | 2 | 0 |
dihedral group:D16 | 7 | 3 | 4 | 3 | 0 |
semidihedral group:SD16 | 8 | 3 | 4 | 3 | 0 |
generalized quaternion group:Q16 | 9 | 3 | 4 | 3 | 0 |
direct product of Z4 and V4 | 10 | 1 | 16 | 0 | 0 |
direct product of D8 and Z2 | 11 | 2 | 8 | 2 | 0 |
direct product of Q8 and Z2 | 12 | 2 | 8 | 2 | 0 |
central product of D8 and Z4 | 13 | 2 | 8 | 2 | 0 |
elementary abelian group:E16 | 14 | 1 | 16 | 0 | 0 |
Here is the lower central series information. Again, notice that the nilpotency class determines the information. Further information: nilpotency class and order determine orders of lower central series members for groups up to prime-fourth order
Group | GAP ID | Nilpotency class | Prime-base logarithm of first lower central series quotient | Prime-base logarithm of second lower central series quotient | Prime-base logarithm of third lower central series quotient |
---|---|---|---|---|---|
cyclic group:Z16 | 1 | 1 | 4 | 0 | 0 |
direct product of Z4 and Z4 | 2 | 1 | 4 | 0 | 0 |
SmallGroup(16,3) | 3 | 2 | 3 | 1 | 0 |
nontrivial semidirect product of Z4 and Z4 | 4 | 2 | 3 | 1 | 0 |
direct product of Z8 and Z2 | 5 | 1 | 4 | 0 | 0 |
M16 | 6 | 2 | 3 | 1 | 0 |
dihedral group:D16 | 7 | 3 | 2 | 1 | 1 |
semidihedral group:SD16 | 8 | 3 | 2 | 1 | 1 |
generalized quaternion group:Q16 | 9 | 3 | 2 | 1 | 1 |
direct product of Z4 and V4 | 10 | 1 | 4 | 0 | 0 |
direct product of D8 and Z2 | 11 | 2 | 3 | 1 | 0 |
direct product of Q8 and Z2 | 12 | 2 | 3 | 1 | 0 |
central product of D8 and Z4 | 13 | 2 | 3 | 1 | 0 |
elementary abelian group:E16 | 14 | 1 | 4 | 0 | 0 |
Here is the upper central series information. Again, notice that the nilpotency class determines the information. Further information: nilpotency class and order determine orders of upper central series members for groups up to prime-fourth order
Group | GAP ID | Nilpotency class | Prime-base logarithm of first upper central series quotient | Prime-base logarithm of second upper central series quotient | Prime-base logarithm of third upper central series quotient |
---|---|---|---|---|---|
cyclic group:Z16 | 1 | 1 | 4 | 0 | 0 |
direct product of Z4 and Z4 | 2 | 1 | 4 | 0 | 0 |
SmallGroup(16,3) | 3 | 2 | 2 | 2 | 0 |
nontrivial semidirect product of Z4 and Z4 | 4 | 2 | 2 | 2 | 0 |
direct product of Z8 and Z2 | 5 | 1 | 4 | 0 | 0 |
M16 | 6 | 2 | 2 | 2 | 0 |
dihedral group:D16 | 7 | 3 | 1 | 1 | 2 |
semidihedral group:SD16 | 8 | 3 | 1 | 1 | 2 |
generalized quaternion group:Q16 | 9 | 3 | 1 | 1 | 2 |
direct product of Z4 and V4 | 10 | 1 | 4 | 0 | 0 |
direct product of D8 and Z2 | 11 | 2 | 2 | 2 | 0 |
direct product of Q8 and Z2 | 12 | 2 | 2 | 2 | 0 |
central product of D8 and Z4 | 13 | 2 | 2 | 2 | 0 |
elementary abelian group:E16 | 14 | 1 | 4 | 0 | 0 |
Group properties
Basic properties
The column headings represent the IDs of the groups:
Property | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
cyclic group | Yes | No | No | No | No | No | No | No | No | No | No | No | No | No |
elementary abelian group | No | No | No | No | No | No | No | No | No | No | No | No | No | Yes |
homocyclic group | Yes | Yes | No | No | No | No | No | No | No | No | No | No | No | Yes |
metacyclic group | Yes | Yes | No | No | Yes | Yes | Yes | Yes | Yes | No | No | No | No | No |
abelian group | Yes | Yes | No | No | Yes | No | No | No | No | Yes | No | No | No | Yes |
Property | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
ambivalent group | No | No | No | No | No | No | Yes | No | Yes | No | Yes | Yes | No | Yes |
rational group | No | No | No | No | No | No | No | No | No | No | Yes | Yes | No | Yes |
rational-representation group | No | No | No | No | No | No | No | No | No | No | Yes | No | No | Yes |
group in which every element is automorphic to its inverse | Yes | Yes | ? | ? | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes |
Property | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
group in which every element is automorphic to its inverse | Yes | Yes | ? | ? | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes |
group in which any two elements generating the same cyclic subgroup are automorphic | Yes | Yes | ? | ? | Yes | ? | Yes | Yes | Yes | Yes | Yes | Yes | ? | Yes |
group of prime power order in which all maximal subgroups are isomorphic | Yes | Yes | No | Yes | No | No | No | No | No | No | No | No | No | Yes |
group of prime power order in which all maximal subgroups are automorphic | Yes | Yes | No | Yes | No | No | No | No | No | No | No | No | No | Yes |
group of prime power order in which every maximal subgroup is isomorph-free | Yes | No | No | No | No | No | No | Yes | No | No | No | No | No | No |
UL-equivalent group | Yes | Yes | No | No | Yes | No | Yes | Yes | Yes | Yes | No | No | No | Yes |
Families and classification
Up to isoclinism
FACTS TO CHECK AGAINST for isoclinic groups (groups with an isoclinism between them):
by definition, isoclinic groups have isomorphic inner automorphism groups and isomorphic derived subgroups, with the isomorphisms compatible.
isoclinic groups have same nilpotency class|isoclinic groups have same derived length | isoclinic groups have same proportions of conjugacy class sizes | isoclinic groups have same proportions of degrees of irreducible representations
The equivalence classes up to being isoclinic were classified by Hall and Senior, and we call them Hall-Senior families.
Up to Hall-Senior genus
Two groups have the same Hall-Senior genus if they are isoclinic and their lattice of normal subgroups are isomorphic to each other.
Genus name | Members | Second part of GAP ID of members | Hall-Senior numbers of members |
---|---|---|---|
elementary abelian group:E16 | 14 | 1 | |
direct product of Z4 and V4 | 10 | 2 | |
direct product of Z4 and Z4 | 2 | 3 | |
direct product of Z8 and Z2 | 5 | 4 | |
cyclic group:Z16 | 1 | 5 | |
direct product of D8 and Z2, direct product of Q8 and Z2 | 11, 12 | 6, 7 | |
central product of D8 and Z4 | 13 | 8 | |
SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4 | 3, 4 | 9, 10 | |
M16 | 6 | 11 | |
dihedral group:D16, semidihedral group:SD16, generalized quaternion group:Q16 | 7, 8, 9 | 12, 13, 14 |
Up to isologism for class two
Under the equivalence relation of being isologic groups with respect to the variety of groups of nilpotency class two, there are two equivalence classes:
Isomorphism class of quotient by second center | Isomorphism class of third member of lower central series | Number of groups | Nilpotency class(es) | Second part of GAP ID of members (sorted ascending) | Hall-Senior numbers of members (sorted ascending) | Smallest order of group isologic to these groups | Stem groups (groups of smallest order) |
---|---|---|---|---|---|---|---|
trivial group | trivial group | 11 | 1,2 | 1,2,3,4,5,6,10,11,12,13,14 | 1-11 | 1 | trivial group |
Klein four-group | cyclic group:Z2 | 3 | 3 | 7,8,9 | 12-14 | 16 | dihedral group:D16, semidihedral group:SD16, generalized quaternion group:Q16 |
Up to isologism for higher class
For any class equal to three or higher, there is a single equivalence class under isologism for that class for groups of order 16, because all groups of order 16 have class at most three and are hence isologic to the trivial group.
Element structure
Further information: element structure of groups of order 16
1-isomorphism
Further information: element structure of groups of order 16#1-isomorphism
There are two pairs of 1-isomorphic groups: M16 (ID: 6) and direct product of Z8 and Z2 (ID: 5) are 1-isomorphic to each other, and central product of D8 and Z4 (ID: 13)and direct product of Z4 and V4 (ID: 10) are 1-isomorphic to each other.
Order statistics
Further information: element structure of groups of order 16#Order statistics
There are many pairs of order statistics-equivalent groups. In addition to the 1-isomorphic pairs, we also have that direct product of Q8 and Z2 (ID: 12) and nontrivial semidirect product of Z4 and Z4 (ID:4) have the same order statistics as direct product of Z4 and Z4 (ID: 2). Also, the group SmallGroup(16,3) (ID: 3)is order statistics-equivalent to direct product of Z4 and V4, but not 1-isomorphic to it.
Conjugacy class structure
Further information: element structure of groups of order 16#Conjugacy class-cum-order statistics
The sizes of conjugacy classes are completely determined by the nilpotency class of the group. Class 1 (abelian) means that all conjugacy classes are of size 1. Class exactly two means that there are four conjugacy classes of size 1 and six conjugacy classes of size 2. Class exactly three means two conjugacy classes of size 1, three of size 2, and two of size 4.
Linear representation theory
Further information: linear representation theory of groups of order 16
Cohomology: piecing together from groups of smaller orders
Central extensions with abelian groups both sides
We list here those groups arising as extensions with base group abelian, quotient group also abelian, and the action trivial.
Automorphism groups
Subgroup structure
Further information: subgroup structure of groups of order 16