Groups of order 64: Difference between revisions

Revision as of 05:17, 24 November 2011

This article gives information about, and links to more details on, groups of order 64
See pages on algebraic structures of order 64 | See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 64. See also more detailed information on specific subtopics through the links:

Information type	Page summarizing information for groups of order 64
element structure (element orders, conjugacy classes, etc.)	element structure of groups of order 64
subgroup structure	subgroup structure of groups of order 64
linear representation theory	linear representation theory of groups of order 64 projective representation theory of groups of order 64 modular representation theory of groups of order 64
endomorphism structure, automorphism structure	endomorphism structure of groups of order 64
group cohomology	group cohomology of groups of order 64

Statistics at a glance

To understand these in a broader context, see
groups of order 2^n|groups of prime-sixth order

Numbers of groups

Since $64 = 2^{6}$ 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	267
Number of abelian groups up to isomorphism	11	Equals the number of unordered integer partitions of $6$ . See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of groups of class exactly two	117
Number of groups of class exactly three	114
Number of groups of class exactly four	22
Number of groups of class exactly five, i.e., maximal class groups	3	classification of finite 2-groups of maximal class. For order $2^{n}, n \geq 4$ , there are exactly three maximal class groups: dihedral, semidihedral, and generalized quaternion. For order 64, the groups are: dihedral group:D64, semidihedral group:SD64, and generalized quaternion group:Q64.

Arithmetic functions

Summary information

Here, the rows are arithmetic functions that take values between $0$ and $6$ , 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 $267$ , which is the total number of groups of order $64$ .

Arithmetic function	Value 0	Value 1	Value 2	Value 3	Value 4	Value 5	Value 6
prime-base logarithm of exponent	0	1	96	137	27	5	1
Frattini length	0	1	94	139	27	5	1
nilpotency class	0	11	117	114	22	3	0
derived length	0	11	256	0	0	0	0
minimum size of generating set	0	1	53	137	68	7	1
rank of a p-group	0	2	54	150	55	5	1
normal rank of a p-group	0	4	87	122	48	5	1
characteristic rank of a p-group

@@ Line 6: / Line 6: @@
 {{quotation|To understand these in a broader context, see<br>[[groups of order 2^n]]<nowiki>|</nowiki>[[groups of prime-sixth order]]}}
+===Numbers of groups===
 Since <math>64= 2^6</math> is a [[prime power]], and [[prime power order implies nilpotent]], all groups of this order are [[nilpotent group]]s.
@@ Line 13: / Line 14: @@
 | Number of groups up to isomorphism || 267 ||
 |-
-| Number of abelian groups up to isomorphism || 11 || Equals the number of [[unordered integer partitions]] of <math>6</math>. See also [[classification of finite abelian groups]].
+| Number of abelian groups up to isomorphism || 11 || Equals the number of [[unordered integer partitions]] of <math>6</math>. {{abelian count explanation}}
 |-
 | Number of groups of class ''exactly'' two || 117 ||
@@ Line 21: / Line 22: @@
 | Number of groups of class ''exactly'' four || 22 ||
 |-
-| Number of groups of class ''exactly'' five, i.e., [[maximal class group]]s || 3 || The dihedral group, semidihedral group, and generalized quaternion group.
+| Number of groups of class ''exactly'' five, i.e., [[maximal class group]]s || 3 || [[classification of finite 2-groups of maximal class]]. For order <math>2^n, n \ge 4</math>, there are exactly three maximal class groups: dihedral, semidihedral, and generalized quaternion. For order 64, the groups are: [[dihedral group:D64]], [[semidihedral group:SD64]], and [[generalized quaternion group:Q64]].
 |}