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C₂²×C₃×#157 #89118

order	432
isoclinism	isoclinic class of #157
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³×C₃

Aliases

C₂²×C₃×#157

Splittings

Fitting, socle	C₂²×C₃³ ⋊ C₄
derived	C₃² ⋊ C₂²×C₃×C₄
Sylow	C₃³ ⋊ C₂²×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₃	#157
commutator, hypocenter	C₃²	C₂²×C₃×C₄
Frattini	C₁	G
Fitting, socle	C₂²×C₃³	C₄
prime radical	C₂²×C₃×C₃⋊D₃	C₂
cosocle	C₃⋊D₃	C₂³×C₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂²×C₃

192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11722722, C₃×#63737, #11722791, …
maximal normal subgroup of	#11722722, C₃×#63737, #11722791, …
maximal quotient of	C₂×C₃×#3787, C₂×C₃×#3788, C₃×#64283, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 765);