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Q₈×C₃⋊Dic₃ #64154

order	288
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₃

Aliases

Q₈×C₃⋊Dic₃

Splittings

C₃² ⋊ C₄×Q₈

Subgroups

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

210 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 18 points:

(1,2,3,4)(14,15)(17,18);(5,7,8,12)(6,9,10,11)(16,17,18);(5,6,8,10)(7,11,12,9)(13,14,15)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Q₈⋊#622, #10591571, #10591572, …
maximal normal subgroup of	Q₈⋊#622, #10591571, #10591572, …
maximal quotient of	C₄⋊C₄×C₃⋊Dic₃, #10592679, #10592683, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 802);