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C₂×#634⋊C₃ #88701

order	432
isoclinism	isoclinic class of #634⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product	C₂×#634⋊C₃
unambiguous semidirect product	(C₂×#634)⋊C₃

Splittings

H(3,𝔽₃) ⋊ C₂×Q₈

Subgroups

	subgroup	G/H
center	C₂²	C₂×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₂×C₃²	C₂³×C₃
Frattini	C₂×C₃	C₂²×C₃×D₃
Fitting	C₂×C₄×H(3,𝔽₃)	C₂
socle	C₂²×C₃	C₂×C₃×D₃
prime radical	C₂²×H(3,𝔽₃)	C₂²
hypercenter	C₂×C₄	(C₃⋊D₃)⋊C₃
hypocenter	C₃²	C₂×C₃×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

149 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11720309, #11720313, #11720321, …
maximal normal subgroup of	#11720309, #11720313, #11720321, …
maximal quotient of	C₄×#634⋊C₃, #64078⋊C₃, #64079⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 348);