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C₃²×#1368 #11768417

order	1152
nilpotency class	4
isoclinism	isoclinic class of #365
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₈ · Schur multiplier (H₂) = C₂×C₃

Aliases

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃²	#106
commutator	C₈	C₂×C₃²×C₈
Frattini, cosocle	C₄×C₈	C₂²×C₃²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₄×C₈ ◁ G

204 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)(7,8,9,11,10,12,13,14)(16,20,21,26)(17,23)(18,25)(22,29,28,27)(24,30);(4,5,6)(15,16,17,20,18,22,23,27,19,21,24,26,25,28,30,29)

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72245532, #72255445
maximal subgroup of	C₃⋊D₃×#1368, C₃²×#12067, #72222453, …
maximal normal subgroup of	C₃⋊D₃×#1368, C₃²×#12067, #72222453, …
maximal quotient of	C₃²×#7099, C₃²×#7457, C₂×C₃²×#1368

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2428);