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C₃×#3774 #88929

order	432
isoclinism	isoclinic class of #3774
automorphism	Aut(G) = C₂×#10596417 (order 1152) · Inn(G) = GO⁺(4,𝔽₂) · Out(G) = C₂×D₄
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

Splittings

C₃³ ⋊ D₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	GO⁺(4,𝔽₂)
commutator, cosocle	C₃⋊Dic₃	C₂²×C₃
Frattini	C₂	C₃×GO⁺(4,𝔽₂)
Fitting, socle	C₂×C₃³	D₄
hypocenter	C₃²	C₃×D₈
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 17 points:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3774, #11724127, #11724132, …
maximal normal subgroup of	D₃×#3774, #11724127, #11724132, …
maximal quotient of	C₃×#63734, C₃×#63735, C₃×#63736, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 576);