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C₂×#3756 #64152

order	288
isoclinism	isoclinic class of #3753
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₃

Aliases

Splittings

C₃² ⋊ C₂×Q₁₆

Subgroups

	subgroup	G/H
center	C₂²	#638
commutator, cosocle	C₃²×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×Q₈	C₂
socle, prime radical	C₂²×C₃²	D₄
hypercenter	C₂×Q₈	C₃⋊D₃
hypocenter	C₃²	C₂×Q₁₆
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10591537, #10591541, #10591575, …
maximal normal subgroup of	#10591537, #10591541, #10591575, …
maximal quotient of	C₂×#63634, C₂×#63637, #10592669, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 800);