Finite Group Databasebeta

← Prev (#6174) Next (#6176) →

C₃×C₉⋊C₈ #6175

order	216
isoclinism	isoclinic class of D₉
automorphism	Aut(G) = C₂³×AGL(1,ℤ/9) (order 432) · Inn(G) = D₉ · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₃×C₈ · Schur multiplier (H₂) = C₁

Aliases

C₃×C₉⋊C₈

Splittings

derived	C₉ ⋊ C₃×C₈
Sylow	C₃×C₉ ⋊ C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₄	D₉
commutator, hypocenter	C₉	C₃×C₈
Frattini	C₃×C₄	C₃×D₃
Fitting	C₃×C₄×C₉	C₂
socle, prime radical	C₂×C₃²	Dic₃
cosocle	C₄×C₉	C₂×C₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₃×C₄

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2,3)(5,7)(6,10)(8,9)(11,12)(13,17,19,14,16,20,15,18);(4,12,7,10,9,8,6,5,11)

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	(C₃⋊D₉)⋊C₈, #88414, D₃×C₉⋊C₈, …
maximal normal subgroup of	(C₃⋊D₉)⋊C₈, #88414, D₃×C₉⋊C₈, …
maximal quotient of	C₃×C₉⋊C₁₆, C₂×C₃×C₉⋊C₈, C₉×C₉⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 12);