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C₂×C₄×D₃ #233

order	48
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = D₃×#120 (order 192) · Inn(G) = D₃ · Out(G) = #120
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

C₂×C₄×D₃

Splittings

Fitting	C₂×C₃×C₄ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂²×C₄

Subgroups

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

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 3 generators on 9 points:

(1,2)(4,5);(3,4,5)(6,7,8,9);(1,2)

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	(C₂×D₃)⋊C₈, C₄²×D₃, #872, …
maximal normal subgroup of	(C₂×D₃)⋊C₈, C₄²×D₃, #872, …
maximal quotient of	C₄×Dic₆, C₄²×D₃, #872, …
automorphism group of	C₃⋊C₁₆, C₅×Dic₃, C₂×C₅×D₃, …
outer automorphism group of	C₂²×C₃×C₅, #504, C₅×#41, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 35);