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C₂²×C₃×Dic₉ #88725

order	432
isoclinism	isoclinic class of D₉
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

C₂²×C₃×Dic₉

Splittings

derived	C₉ ⋊ C₂²×C₃×C₄
Sylow	C₃×C₉ ⋊ C₂²×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂³×C₃	D₉
commutator, hypocenter	C₉	C₂²×C₃×C₄
Frattini	C₂×C₃	C₂²×C₃×D₃
Fitting	C₂³×C₃×C₉	C₂
socle, prime radical	C₂³×C₃²	D₃
cosocle	C₂×C₉	C₂³×C₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₂³×C₃

194 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 20 points:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11720331, C₃×C₉⋊#95, #11721241, …
maximal normal subgroup of	#11720331, C₃×C₉⋊#95, #11721241, …
maximal quotient of	C₂²×C₃×C₉⋊C₈, C₂×C₃×#3667, C₂×C₃×C₄×Dic₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 372);