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C₃²×#1311 #11768360

order	1152
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₃²×C₈² · Schur multiplier (H₂) = C₃×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₃²×C₄×C₈	C₂²
commutator	C₂	C₃²×C₈²
Frattini, cosocle	C₄×C₈	C₂²×C₃²
socle, prime radical	C₂²×C₃²	C₄×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃²×C₄×C₈ ◁ G

288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 1152

A minimal faithful permutation representation — 2 generators on 30 points:

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

Conjugacy classes & character table

720 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₃⋊D₃×#1311, #72245300, #72245362, …
maximal subgroup of	C₃²×#7330, C₃⋊D₃×#1311, #72245300, …
maximal normal subgroup of	C₃²×#7330, C₃⋊D₃×#1311, #72245300, …
maximal quotient of	C₃²×#7330

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2371);