C₂²×#87522 #72342883

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂²×#87522

Splittings

Fitting	C₂²×#585 ⋊ D₃
derived	#997 ⋊ C₂⁴

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#1020
commutator, cosocle, hypocenter	#997	C₂⁴
Frattini	C₂	C₂³×#1020
Fitting	C₂²×#585	D₃
hypercenter	C₂²×C₄	#1020
LCS	G ▷ #997
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

13241 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 13241 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2)(3,4)(5,7,6,10)(8,18,16,17,12,20,11,14,15,13,9,19);(3,4)(5,15,6,16)(7,20,10,19)(8,12,11,9)(13,18,17,14);(7,10)(8,15)(9,12)(11,16)(13,19)(17,20);(1,2)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,16)(19,20)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#11714174, #72342894, C₂×#72215707, …
maximal normal subgroup of	C₂²×#11714174, #72342894, C₂×#72215707, …
maximal quotient of	C₂×#72245579

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2)(3,4)(5,7,6,10)(8,18,16,17,12,20,11,14,15,13,9,19), (3,4)(5,15,6,16)(7,20,10,19)(8,12,11,9)(13,18,17,14), (7,10)(8,15)(9,12)(11,16)(13,19)(17,20), (1,2)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,16)(19,20));