C₂²×#10624749 #72228397

order	2916
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃³ · Schur multiplier (H₂) = C₂×C₃⁵

Aliases

direct product

C₂²×#10624749

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₂²×C₃³
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃³ ◁ G

1220 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 1220 conjugacy classes of subgroups — too many to draw. Only its 230 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) = 31
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 3 generators on 31 points:

(1,2)(5,6,9)(7,10,17)(8,12,18)(11,21,16)(13,22,29)(14,20,25)(15,19,26)(23,28,30)(24,27,31);(3,4)(7,13,15)(10,19,22)(20,27,28)(25,30,31);(5,7,14,8,15,24,16,13,23)(6,10,20,12,19,27,11,22,28)(9,17,25,18,26,31,21,29,30)

Conjugacy classes & character table

228 conjugacy classes — view the full character table.

Relationships

derived subgroup of	A₄×#13091120
Fitting subgroup of	A₄×#10624749, C₂²×#13091120, A₄×#13091120, …
maximal subgroup of	A₄×#10624749, C₂²×#13091120, D₄×#10624749, …
maximal normal subgroup of	A₄×#10624749, C₂²×#13091120, D₄×#10624749, …
maximal quotient of	D₄×#10624749

GAP

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

gap> G := Group((1,2)(5,6,9)(7,10,17)(8,12,18)(11,21,16)(13,22,29)(14,20,25)(15,19,26)(23,28,30)(24,27,31), (3,4)(7,13,15)(10,19,22)(20,27,28)(25,30,31), (5,7,14,8,15,24,16,13,23)(6,10,20,12,19,27,11,22,28)(9,17,25,18,26,31,21,29,30));