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C₂×D₃×C₈ #899

order	96
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂×D₃×#120 (order 384) · Inn(G) = D₃ · Out(G) = C₂×#120
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

C₂×D₃×C₈

Splittings

Fitting	C₂×C₃×C₈ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂²×C₈

Subgroups

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

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 13 points:

(1,2)(4,5);(3,4,5)(6,7,8,10,9,11,12,13);(1,2)

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	(C₂×D₃)⋊C₁₆, C₄×D₃×C₈, C₈×D₁₂, …
maximal normal subgroup of	(C₂×D₃)⋊C₁₆, C₄×D₃×C₈, C₈×D₁₂, …
maximal quotient of	C₈×Dic₆, C₄×D₃×C₈, #4685, …
automorphism group of	C₃⋊C₃₂
outer automorphism group of	#2258, #33982, #33984

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 106);