C₃²×#4183⋊C₃ #72244459

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃³

Aliases

direct product

C₃²×#4183⋊C₃

Splittings

Fitting, Sylow	C₃²×#6909 ⋊ C₂
derived	AΣL(1,𝔽₂₇) ⋊ C₂×C₃³

Subgroups

	subgroup	G/H
center, hypercenter	C₃²	#4183⋊C₃
commutator, cosocle, hypocenter	AΣL(1,𝔽₂₇)	C₂×C₃³
Frattini	C₃²	C₃³×C₃⋊D₃
Fitting	C₃²×#6909	C₂
socle	C₃³	C₃×#272
LCS	G ▷ AΣL(1,𝔽₂₇)
UCS	C₁ ◁ C₃²

5944 conjugacy classes of subgroups; only the 386 normal subgroups are shown.

normal subgroups only

This group has 5944 conjugacy classes of subgroups — too many to draw. Only its 386 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) = 33
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 3 generators on 33 points:

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

Conjugacy classes & character table

279 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of

C₃²×#13091120

GAP

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

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