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C₃×C₄×#157 #88983

order	432
isoclinism	isoclinic class of #157
automorphism	?
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₃×C₄

Aliases

C₃×C₄×#157

Splittings

Fitting	C₃³×C₄ ⋊ C₄
derived	C₃² ⋊ C₃×C₄²
Sylow	C₃³ ⋊ C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₄	#157
commutator, hypocenter	C₃²	C₃×C₄²
Frattini	C₂	C₂×C₃×#157
Fitting	C₃³×C₄	C₄
socle	C₂×C₃³	C₂×C₄
prime radical	C₂×C₃×C₃⋊D₃	C₂²
cosocle	C₂×C₃⋊D₃	C₂²×C₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₃×C₄

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊C₈×#157, #11722704, #11722711, …
maximal normal subgroup of	C₃⋊C₈×#157, #11722704, #11722711, …
maximal quotient of	C₃×C₈×#157, C₃×#63767, C₃×C₄×#622, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 630);