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C₃×#3775 #88930

order	432
isoclinism	isoclinic class of #3774
automorphism	Aut(G) = C₂³×GO⁺(4,𝔽₂) (order 576) · Inn(G) = GO⁺(4,𝔽₂) · Out(G) = C₂³
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

Splittings

C₃³ ⋊ SD₁₆

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
prime radical	C₃×#625	C₂
hypocenter	C₃²	C₃×SD₁₆
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃

84 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)(10,15,13,11)(12,14,16,17);(4,5)(6,9)(7,8)(11,14)(12,16)(15,17)

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3775, #11724129, #11724133, …
maximal normal subgroup of	D₃×#3775, #11724129, #11724133, …
maximal quotient of	C₃×#63739, C₃×#63740, C₃×#63742, …
outer automorphism group of	#6898

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 577);