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C₃×C₉⋊C₁₆ #88381

order	432
isoclinism	isoclinic class of D₉
automorphism	Aut(G) = C₂²×C₄×AGL(1,ℤ/9) (order 864) · Inn(G) = D₉ · Out(G) = C₂²×C₃×C₄
homology	Abelianization (H₁) = C₃×C₁₆ · Schur multiplier (H₂) = C₁

Aliases

C₃×C₉⋊C₁₆

Splittings

derived	C₉ ⋊ C₃×C₁₆
Sylow	C₃×C₉ ⋊ C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₈	D₉
commutator, hypocenter	C₉	C₃×C₁₆
Frattini	C₃×C₈	C₃×D₃
Fitting	C₃×C₈×C₉	C₂
socle, prime radical	C₂×C₃²	C₃⋊C₈
cosocle	C₈×C₉	C₂×C₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₃×C₈

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 28 points:

(1,2,3)(5,7)(6,10)(8,9)(11,12)(13,18,21,20,23,28,15,19,17,25,27,14,16,24,22,26);(4,12,7,10,9,8,6,5,11)

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	(C₃⋊D₉)⋊C₁₆, #11720231, D₃×C₉⋊C₁₆, …
maximal normal subgroup of	(C₃⋊D₉)⋊C₁₆, #11720231, D₃×C₉⋊C₁₆, …
maximal quotient of	C₃×C₉⋊C₃₂, C₂×C₃×C₉⋊C₁₆, C₉×C₉⋊C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 28);