Finite Group Databasebeta

← Prev (#88500) Next (#88502) →

C₃×#3672 #88501

order	432
isoclinism	isoclinic class of #3672
automorphism	Aut(G) = C₂²×D₄×AGL(1,ℤ/9) (order 1728) · Inn(G) = #611 · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

Splittings

C₃×C₉ ⋊ SD₁₆

Subgroups

	subgroup	G/H
center	C₂×C₃	#611
commutator, cosocle	C₄×C₉	C₂²×C₃
Frattini	C₃×C₄	C₂×C₃×D₃
Fitting	C₃×D₄×C₉	C₂
socle	C₂×C₃²	#67
prime radical	C₃²×D₄	D₃
hypercenter	C₃×D₄	D₉
hypocenter	C₉	C₃×SD₁₆
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×D₄

76 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)(4,5,8,10,12,11,6,9,7)(14,17)(15,20)(18,19);(5,7)(6,10)(8,9)(11,12)(13,19,16,14)(15,18,20,17)

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3672, #11721165, #11721170, …
maximal normal subgroup of	D₃×#3672, #11721165, #11721170, …
maximal quotient of	C₃×#63367, C₃×#63368, C₃×#63392, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 148);