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C₂×#3832 #64363

order	288
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₃

Aliases

Splittings

Fitting	C₂×C₃²×Q₈ ⋊ C₂
Sylow	C₃² ⋊ C₂×#41

Subgroups

	subgroup	G/H
center	C₂²	C₂²×C₃⋊D₃
commutator, cosocle	C₂×C₃²	C₂⁴
Frattini	C₂	C₂³×C₃⋊D₃
Fitting	C₂×C₃²×Q₈	C₂
socle	C₂²×C₃²	C₂³
hypercenter	C₂×Q₈	C₃⋊D₃
hypocenter	C₃²	C₂×#41
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 16 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10591554, #10591560, #10591593, …
maximal normal subgroup of	#10591554, #10591560, #10591593, …
maximal quotient of	C₂×#64099, C₂×#64101, C₂×#64102, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 1011);