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C₂²×#624 #64324

order	288
isoclinism	isoclinic class of D₃²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂²×#624

Splittings

C₃² ⋊ C₂³×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂³	D₃²
commutator, hypocenter	C₃²	C₂³×C₄
Frattini	C₂	C₂²×D₃²
Fitting, socle	C₂³×C₃²	C₂²
prime radical	C₂³×C₃⋊D₃	C₂
cosocle	C₂×C₃²	C₂⁴
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂³

539 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 4 generators on 14 points:

(1,2)(3,4)(6,7)(8,9,10)(11,13,14,12);(5,6,7)(9,10)(11,12,14,13);(3,4);(1,2)

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10591233, #10591275, #10591367, …
maximal normal subgroup of	#10591233, #10591275, #10591367, …
maximal quotient of	C₂²×#3710, C₂×#63817, C₂×#63818, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 972);