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C₃×#627 #6286

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) = C₁

Aliases

Splittings

C₃³ ⋊ Q₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃²
commutator, cosocle	C₂×C₃²	C₂²×C₃
Frattini	C₂	C₃×D₃²
Fitting, socle, prime radical	C₂×C₃³	C₂²
hypocenter	C₃²	C₃×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃

70 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:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#3775, C₃×#3776, #88936, …
maximal normal subgroup of	C₃×#3775, C₃×#3776, #88936, …
maximal quotient of	C₃×Dic₃⋊Dic₃, C₃×#3724, C₂×C₃×#627, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 123);