Finite Group Databasebeta

← Prev (#6284) Next (#6286) →

C₃×#626 #6285

order	216
isoclinism	isoclinic class of #625
automorphism	Aut(G) = C₂³×D₃² (order 288) · Inn(G) = D₃² · Out(G) = C₂³
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:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#626, #88953, #88954, …
maximal normal subgroup of	D₃×#626, #88953, #88954, …
maximal quotient of	C₃×#3714, C₃×#3716, C₃×#3717, …
outer automorphism group of	#6896

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 122);