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Q₈×C₄²⋊C₃ #73163

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₄

Aliases

Q₈×C₄²⋊C₃

Splittings

C₄²×Q₈ ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂	C₂²×C₄²⋊C₃
commutator, cosocle	C₂×C₄²	C₂²×C₃
Frattini, socle	C₂³	C₂²×A₄
Fitting	C₄²×Q₈	C₃
prime radical	C₂×C₄²⋊C₃	C₂²
hypercenter	Q₈	C₄²⋊C₃
hypocenter	C₄²	C₃×Q₈
LCS	G ▷ C₂×C₄² ▷ C₄²
UCS	C₁ ◁ C₂ ◁ Q₈

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 20 points:

(1,3,4,8)(2,5,6,7)(9,10,13)(11,14,18)(12,15,17)(16,19,20);(1,2,4,6)(3,7,8,5)(9,16,12,11)(13,18,17,20)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	SD₁₆×C₄²⋊C₃, Q₁₆×C₄²⋊C₃, #33079⋊C₃, …
maximal normal subgroup of	SD₁₆×C₄²⋊C₃, Q₁₆×C₄²⋊C₃, #33079⋊C₃, …
maximal quotient of	C₄⋊C₄×C₄²⋊C₃, Q₈×#95⋊C₃, #13812⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5739);