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Q₈×C₃⋊C₁₆ #70948

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

Q₈×C₃⋊C₁₆

Splittings

C₃ ⋊ Q₈×C₁₆

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃
commutator	C₂×C₃	C₂²×C₁₆
Fitting	C₃×C₈×Q₈	C₂
socle, prime radical	C₂²×C₃	C₂²×C₈
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₈×Q₈	D₃
hypocenter	C₃	Q₈×C₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 27 points:

(2,3)(12,13,14,17,15,18,20,23,16,19,21,24,22,25,26,27);(1,2,3)(4,6,7,11)(5,8,9,10);(4,5,7,9)(6,10,11,8)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10706950, SD₁₆×C₃⋊C₁₆, #10706964, …
maximal normal subgroup of	#10706950, SD₁₆×C₃⋊C₁₆, #10706964, …
maximal quotient of	C₄⋊C₄×C₃⋊C₁₆, #10712804, #10713188, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 3524);