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C₂×#3745 #64112

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

Aliases

Splittings

Fitting	C₂×C₃²×C₈ ⋊ C₂
Sylow	C₃² ⋊ C₂×D₈

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	C₂²×C₃²	D₄
hypercenter	C₂×C₈	C₃⋊D₃
hypocenter	C₃²	C₂×D₈
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 16 points:

(3,4,5)(9,10,14,16,12,15,11,13);(1,2)(4,5)(7,8)(10,13)(11,14)(15,16);(1,2)(6,7,8)

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10589116, #10589128, #10589357, …
maximal normal subgroup of	#10589116, #10589128, #10589357, …
maximal quotient of	#10592381, C₄×#3745, #10592393, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 760);