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C₂×#6229 #88737

order	432
isoclinism	isoclinic class of #6229
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

H(3,𝔽₃) ⋊ C₂×Q₈

Subgroups

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

209 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:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11720326, #11720327, #11720532, …
maximal normal subgroup of	#11720326, #11720327, #11720532, …
maximal quotient of	C₄×#6229, #11721715, #11721716, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 384);