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Q₈×C₃⋊C₈ #5023

order	192
automorphism	Aut(G) = C₂³×D₃×#1020 (order 4608) · Inn(G) = C₂²×D₃ · Out(G) = C₂³×S₄
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

Q₈×C₃⋊C₈

Splittings

C₃ ⋊ C₈×Q₈

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃
commutator	C₂×C₃	C₂²×C₈
Fitting	C₃×C₄×Q₈	C₂
socle, prime radical	C₂²×C₃	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₄×Q₈	D₃
hypocenter	C₃	C₈×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#68212, #68213, #68214, …
maximal normal subgroup of	#68212, #68213, #68214, …
maximal quotient of	C₄⋊C₄×C₃⋊C₈, #69482, #69484, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 582);