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C₃²×#1455 #11768504

order	1152
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₄² · Schur multiplier (H₂) = C₃×C₈

Aliases

Subgroups

	subgroup	G/H
center	C₃²×C₄²	C₂³
commutator	C₂²	C₂×C₃²×C₄²
Frattini, cosocle	C₄²	C₂³×C₃²
socle	C₂²×C₃²	C₂×C₄²
prime radical	C₂³×C₃²	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₃²×C₄² ◁ G

642 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 22 points:

(1,2,3)(7,8,9,12,10,11,13,14);(4,5,6)(15,16,17,20,18,19,21,22);(8,11)(12,14)(16,19)(20,22)

Conjugacy classes & character table

504 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₃×D₃×#1455, C₃×#10784428
maximal subgroup of	C₃⋊D₃×#1455, C₃²×#19745, #72214093, …
maximal normal subgroup of	C₃⋊D₃×#1455, C₃²×#19745, #72214093, …
maximal quotient of	3₋¹⁺²×#1455

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2515);