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C₂×#3746 #64114

order	288
isoclinism	isoclinic class of #3744
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₃

Aliases

Splittings

C₃² ⋊ C₂×Q₁₆

Subgroups

	subgroup	G/H
center	C₂²	C₄⋊(C₃⋊D₃)
commutator, cosocle	C₃²×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×C₈	C₂
socle, prime radical	C₂²×C₃²	D₄
hypercenter	C₂×C₈	C₃⋊D₃
hypocenter	C₃²	C₂×Q₁₆
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 24 points:

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

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10589119, #10589131, #10589358, …
maximal normal subgroup of	#10589119, #10589131, #10589358, …
maximal quotient of	#10592380, #10592381, C₄×#3746, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 762);