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C₃×#625 #6284

order	216
isoclinism	isoclinic class of #625
automorphism	Aut(G) = C₂×#64241 (order 576) · Inn(G) = D₃² · Out(G) = C₂×D₄
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

Splittings

C₃³ ⋊ D₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃²
commutator, cosocle	C₂×C₃²	C₂²×C₃
Frattini	C₂	C₃×D₃²
Fitting, socle	C₂×C₃³	C₂²
hypocenter	C₃²	C₃×D₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃

94 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#3774, C₃×#3775, #88935, …
maximal normal subgroup of	C₃×#3774, C₃×#3775, #88935, …
maximal quotient of	C₃×#3713, C₃×#3715, C₃×#3718, …
outer automorphism group of	#6856

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 121);