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C₃×Q₈⋊C₈ #4573

order	192
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₂⁵×D₄ (order 256) · Inn(G) = D₄ · Out(G) = C₂⁵
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂

Aliases

C₃×Q₈⋊C₈

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	D₄
commutator	C₄	C₂×C₃×C₈
Frattini, cosocle	C₄²	C₂²×C₃
socle, prime radical	C₂²×C₃	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄² ◁ G

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#68212, #68213, #68214, …
maximal subgroup of	#68212, #68213, #68214, …
maximal normal subgroup of	#68212, #68213, #68214, …
maximal quotient of	C₃×#1271, C₃×#1272, C₃×#1279, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 132);