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C₂²×C₃⋊#157 #89119

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

Aliases

C₂²×C₃⋊#157

Splittings

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

Subgroups

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

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:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11722722, #11722769, C₃⋊#63747, …
maximal normal subgroup of	#11722722, #11722769, C₃⋊#63747, …
maximal quotient of	C₂×#88988, C₂×#88989, #11724511, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 766);