C₂×#72229320 #72276253

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#72229320

Splittings

derived

C₂²⋊(C₂²⋊C₂₇) ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72229320
commutator, cosocle, hypocenter	C₂²⋊(C₂²⋊C₂₇)	C₂²×C₃
Frattini	C₉	C₂×C₃×#1020
Fitting	C₂⁵×C₃×C₉	D₃
socle	C₂⁵×C₃	C₃×D₉
LCS	G ▷ C₂²⋊(C₂²⋊C₂₇)
UCS	C₁ ◁ C₂

1588 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 1588 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 37
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 2 generators on 37 points:

(3,6,5,4)(7,9,10,8)(12,24,19,15,16,33)(13,20)(14,23)(17,36,25,27,28,31)(18,29,32,30,26,37)(21,34)(22,35);(1,2)(4,5,6)(8,9,10)(11,18,24,22,36,25,20,12,29,14,32,33,34,27,17,21,19,37,23,26,15,13,31,28,35,16,30)

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

GAP

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

gap> G := Group((3,6,5,4)(7,9,10,8)(12,24,19,15,16,33)(13,20)(14,23)(17,36,25,27,28,31)(18,29,32,30,26,37)(21,34)(22,35), (1,2)(4,5,6)(8,9,10)(11,18,24,22,36,25,20,12,29,14,32,33,34,27,17,21,19,37,23,26,15,13,31,28,35,16,30));