# Groups of order 32

This article gives information about, and links to more details on, groups of order 32

See pages on algebraic structures of order 32| See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 32. See also more detailed information on specific subtopics through the links:

Information type | Page summarizing information for groups of order 32 |
---|---|

element structure (element orders, conjugacy classes, etc.) | element structure of groups of order 32 |

subgroup structure | subgroup structure of groups of order 32 |

linear representation theory | linear representation theory of groups of order 32 projective representation theory of groups of order 32 modular representation theory of groups of order 32 |

endomorphism structure, automorphism structure | endomorphism structure of groups of order 32 |

group cohomology | group cohomology of groups of order 32 |

## Statistics at a glance

### Numbers of groups

To understand these in a broader context, see

groups of order 2^n|groups of prime-fifth order

Since is a prime power, and prime power order implies nilpotent, all groups of this order are nilpotent groups.

Quantity | Value | Explanation |
---|---|---|

Number of groups up to isomorphism | 51 | |

Number of abelian groups up to isomorphism | 7 | Equals the number of unordered integer partitions of 5. See classification of finite abelian groups and structure theorem for finitely generated abelian groups. |

Number of groups of class exactly two up to isomorphism |
26 | |

Number of groups of class exactly three up to isomorphism |
15 | |

Number of groups of class exactly four up to isomorphism, 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 32, the groups are: dihedral group:D32, semidihedral group:SD32, and generalized quaternion group:Q32. |

### Numbers of equivalence classes of groups

Equivalence relation on groups | Number of equivalence classes of groups of order 32 | Sizes of equivalence classes, i.e., number of isomorphism classes of groups within each equivalence class (should add up to 51) | More information |
---|---|---|---|

isoclinic groups (i.e., Hall-Senior families) | 8 | 7, 15, 10, 9, 2, 2, 3, 3 | #Up to isoclinism, see also classification of groups of order 32 |

isologic groups with respect to nilpotency class two | 3 | 33, 15, 3 | #Up to isologism for class two. This is a coarser equivalence relation than being isoclinic. |

isologic groups with respect to nilpotency class three | 2 | 48, 3 | #Up to isologism for class three. This is a coarser equivalence relation than being isologic with respect to nilpotency class two. |

having the same conjugacy class size statistics | 6 | 7,15,19,2,5,3 | Element structure of groups of order 32#Conjugacy class sizes. Note that isoclinic groups of the same order have the same conjugacy class size statistics, so this is a coarser equivalence relation than being isoclinic. |

having the same degrees of irreducible representations | 6 | 7,15,19,2,5,3 | See Linear representation theory of groups of order 32#Degrees of irreducible representations. Note that for order 32, the degrees of irreducible representations and the conjugacy class size statistics determine each other, but this breaks down for higher orders. Also, note that this is a coarser equivalence relation than being isoclinic. |

1-isomorphic groups | 38 | 1 (29 times), 2 (6 times), 3 (2 times), 4 (1 time) | Element structure of groups of order 32#1-isomorphism |

## The list

Group | Second part of GAP ID (GAP ID is (32,second part)) | Hall-Senior number (among groups of order 32) | Hall-Senior symbol | Nilpotency class | Probability in cohomology tree probability distribution (proper fraction) | Probability in cohomology tree probability distribution (as numerical value) |
---|---|---|---|---|---|---|

Cyclic group:Z32 | 1 | 7 | 1 | 1/16 | 0.0625 | |

SmallGroup(32,2) | 2 | 18 | 2 | 59/2048 | 0.0288 | |

Direct product of Z8 and Z4 | 3 | 5 | 1 | 51/1024 | 0.0498 | |

Semidirect product of Z8 and Z4 of M-type | 4 | 19 | 2 | 49/1024 | 0.0479 | |

SmallGroup(32,5) | 5 | 20 | 2 | 71/1024 | 0.0693 | |

Faithful semidirect product of E8 and Z4 | 6 | 46 | 3 | 13/1024 | 0.0127 | |

SmallGroup(32,7) | 7 | 47 | 3 | 13/2048 | 0.0063 | |

SmallGroup(32,8) | 8 | 48 | 3 | 13/2048 | 0.0063 | |

SmallGroup(32,9) | 9 | 27 | 3 | 31/1024 | 0.0303 | |

SmallGroup(32,10) | 10 | 28 | 3 | 37/1024 | 0.0361 | |

Wreath product of Z4 and Z2 | 11 | 31 | 3 | 13/512 | 0.0254 | |

SmallGroup(32,12) | 12 | 21 | 2 | 45/512 | 0.0879 | |

Semidirect product of Z8 and Z4 of semidihedral type | 13 | 30 | 3 | 7/256 | 0.0273 | |

Semidirect product of Z8 and Z4 of dihedral type | 14 | 29 | 3 | 25/1024 | 0.0244 | |

SmallGroup(32,15) | 15 | 32 | 3 | 1/32 | 0.0313 | |

Direct product of Z16 and Z2 | 16 | 6 | 1 | 31/256 | 0.1211 | |

M32 | 17 | 22 | 2 | 15/256 | 0.0586 | |

Dihedral group:D32 | 18 | 49 | 4 | 3/1024 | 0.0029 | |

Semidihedral group:SD32 | 19 | 50 | 4 | 3/512 | 0.0059 | |

Generalized quaternion group:Q32 | 20 | 51 | 4 | 3/1024 | 0.0029 | |

Direct product of Z4 and Z4 and Z2 | 21 | 3 | 1 | 637/65536 | 0.0097 | |

Direct product of SmallGroup(16,3) and Z2 | 22 | 11 | 2 | 695/65536 | 0.0106 | |

Direct product of SmallGroup(16,4) and Z2 | 23 | 12 | 2 | 349/16384 | 0.0213 | |

SmallGroup(32,24) | 24 | 16 | 2 | 273/32768 | 0.0083 | |

Direct product of D8 and Z4 | 25 | 14 | 2 | 69/4096 | 0.0168 | |

Direct product of Q8 and Z4 | 26 | 15 | 2 | 123/16384 | 0.0075 | |

SmallGroup(32,27) | 27 | 33 | 2 | 45/16384 | 0.0027 | |

SmallGroup(32,28) | 28 | 36 | 2 | 33/4096 | 0.0081 | |

SmallGroup(32,29) | 29 | 37 | 2 | 225/16384 | 0.0137 | |

SmallGroup(32,30) | 30 | 38 | 2 | 129/16384 | 0.0079 | |

SmallGroup(32,31) | 31 | 39 | 2 | 129/32768 | 0.0039 | |

SmallGroup(32,32) | 32 | 40 | 2 | 111/16384 | 0.0068 | |

SmallGroup(32,33) | 33 | 41 | 2 | 21/4096 | 0.0051 | |

Generalized dihedral group for direct product of Z4 and Z4 | 34 | 34 | 2 | 45/65536 | 0.0007 | |

SmallGroup(32,35) | 35 | 35 | 2 | 321/65536 | 0.0049 | |

Direct product of Z8 and V4 | 36 | 4 | 1 | 543/16384 | 0.0331 | |

Direct product of M16 and Z2 | 37 | 13 | 2 | 637/16384 | 0.0389 | |

Central product of D8 and Z8 | 38 | 17 | 2 | 63/4096 | 0.0154 | |

Direct product of D16 and Z2 | 39 | 23 | 3 | 141/32768 | 0.0043 | |

Direct product of SD16 and Z2 | 40 | 24 | 3 | 237/16384 | 0.0145 | |

Direct product of Q16 and Z2 | 41 | 25 | 3 | 237/32768 | 0.0072 | |

Central product of D16 and Z4 | 42 | 26 | 3 | 45/8192 | 0.0055 | |

Holomorph of Z8 | 43 | 44 | 3 | 45/8192 | 0.0055 | |

SmallGroup(32,44) | 44 | 45 | 3 | 45/8192 | 0.0055 | |

Direct product of E8 and Z4 | 45 | 2 | 1 | 1023/1048576 | 0.0010 | |

Direct product of D8 and V4 | 46 | 8 | 2 | 825/1048576 | 0.0008 | |

Direct product of Q8 and V4 | 47 | 9 | 2 | 771/1048576 | 0.0007 | |

Direct product of SmallGroup(16,13) and Z2 | 48 | 10 | 2 | 329/262144 | 0.0013 | |

Inner holomorph of D8 | 49 | 42 | 2 | 35/131072 | 0.0003 | |

Central product of D8 and Q8 | 50 | 43 | 2 | 21/131072 | 0.0002 | |

Elementary abelian group:E32 | 51 | 1 | 1 | 1/1048576 | 0.0000 |

## 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 . To view a list of all groups, click on the value in the cell and the list of all groups with GAP IDs appears.

Arithmetic function | Value 0 | Value 1 | Value 2 | Value 3 | Value 4 | Value 5 | Mean (with equal weighting on all groups) | Mean (with weighting by cohomology tree probability distribution) |
---|---|---|---|---|---|---|---|---|

prime-base logarithm of exponent | 0 | 1 | 23 | 21 | 5 | 1 | 2.6471 | 3.1426 |

Frattini length | 0 | 1 | 23 | 21 | 5 | 1 | 2.6471 | 3.1426 |

nilpotency class | 0 | 7 | 26 | 15 | 3 | 0 | 2.2745 | 1.9889 |

derived length | 0 | 7 | 44 | 0 | 0 | 0 | 1.8627 | 1.7228 |

minimum size of generating set (sometimes called rank, though it differs from rank of a p-group as used below) |
0 | 1 | 19 | 24 | 6 | 1 | 2.7451 | 2.2039 |

rank of a p-group | 0 | 2 | 21 | 23 | 4 | 1 | 2.6275 | 2.3064 |

normal rank of a p-group | 0 | 4 | 23 | 19 | 4 | 1 | 2.5098 | 2.2431 |

characteristic rank of a p-group | 0 | 7 | 26 | 14 | 3 | 1 | 2.3137 | 2.1972 |

## Families and classification

### Up to isoclinism

FACTS TO CHECK AGAINSTfor 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 information below collects groups based on the equivalence relation of being isoclinic groups. The equivalence classes are also called Hall-Senior families.

Family name | Isomorphism class of inner automorphism group | Isomorphism class of derived subgroup | Number of members | Nilpotency class | Members | Second part of GAP ID of members (sorted ascending) | Hall-Senior numbers of members (sorted ascending) |
---|---|---|---|---|---|---|---|

trivial group | trivial group | 7 | 1 | all abelian groups of order 32: [SHOW MORE] | 1,3,16,21,36,45,51 | 1-7 | |

Klein four-group | cyclic group:Z2 | 15 | 2 | [SHOW MORE] | 2,4,5,12,17,22,23,24,25,26,37,38,46,47,48 | 8-22 | |

dihedral group:D8 | cyclic group:Z4 | 10 | 3 | [SHOW MORE] | 9,10,11,13,14,15,39,40,41,42 | 23-32 | |

elementary abelian group:E8 | Klein four-group | 9 | 2 | [SHOW MORE] | 27-35 | 33-41 | |

elementary abelian group:E16 | cyclic group:Z2 | 2 | 2 | inner holomorph of D8, central product of D8 and Q8 | 49, 50 | 42, 43 | |

direct product of D8 and Z2 | cyclic group:Z4 | 2 | 3 | holomorph of Z8, SmallGroup(32,44) | 43,44 | 44,45 | |

SmallGroup(16,3) | Klein four-group | 3 | 3 | [SHOW MORE] | 6-8 | 46-48 | |

dihedral group:D16 | cyclic group:Z8 | 3 | 4 | [SHOW MORE] | 18-20 | 49-51 | |

Total (8 rows) | -- | -- | 51 | -- | -- | -- | -- |

### Up to Hall-Senior genus

### 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, the equivalence classes are as follows (*the table is incomplete*):

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 | 33 | 1,2 | 1-5,12,16,17,21-38,45-51 | 1-22,33-43 | 1 | trivial group |

Klein four-group | cyclic group:Z2 | 15 | 3 | 6-11, 13-15, 39-44 | 23-32, 44-48 | 16 | dihedral group:D16, semidihedral group:SD16, generalized quaternion group:Q16 |

dihedral group:D8 | cyclic group:Z4 | 3 | 4 | 18-20 | 49-51 | 32 | dihedral group:D32, semidihedral group:SD32, generalized quaternion group:Q32 |

-- (3 rows) | -- | 51 | -- | -- | -- | -- | -- |

### Up to isologism for class three

Under the equivalence relation of being isologic groups with respect to the variety of groups of class at most three, there are two equivalence classes:

Isomorphism class of quotient by third center | Isomorphism class of fourth 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 | 48 | 1,2,3 | 1-17,21-51 | 1-48 | 1 | trivial group |

Klein four-group | cyclic group:Z2 | 3 | 4 | 18-20 | 49-51 | 32 | dihedral group:D32, semidihedral group:SD32, generalized quaternion group:Q32 |

-- (2 rows) | -- | 51 | -- | -- | -- | -- | -- |

### Up to isologism for higher class

For class four or higher, all groups of order 32 are isologic to each other.

## Element structure

`Further information: element structure of groups of order 32`

## Subgroup structure

`Further information: subgroup structure of groups of order 32`

## Linear representation theory

`Further information: linear representation theory of groups of order 32`

## References

- First complete published classification:
*The regular substitution groups whose orders are less than 48*by G. A. Miller, Quarterly Journal of Mathematics, Volume 28, Page 232 - 284(Year 1896):^{}^{More info} - Detailed information about the groups:
*The groups of order ()*by Marshall Hall and James Kuhn Senior. Reviewed on MathScinet^{More info}

## GAP implementation

The order 32 is part of GAP's SmallGroup library. Hence, any group of order 32 can be constructed using the SmallGroup function by specifying its group ID. Also, IdGroup is available, so the group ID of any group of this order can be queried.

Further, the collection of all groups of order 32 can be accessed as a list using GAP's AllSmallGroups function.

Here is GAP's summary information about how it stores groups of this order, accessed using GAP's SmallGroupsInformation function:

gap> SmallGroupsInformation(32); There are 51 groups of order 32. They are sorted by their ranks. 1 is cyclic. 2 - 20 have rank 2. 21 - 44 have rank 3. 45 - 50 have rank 4. 51 is elementary abelian. For the selection functions the values of the following attributes are precomputed and stored: IsAbelian, PClassPGroup, RankPGroup, FrattinifactorSize and FrattinifactorId. This size belongs to layer 2 of the SmallGroups library. IdSmallGroup is available for this size.