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C₂×C₄⋊#272 #88739

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

Aliases

C₂×C₄⋊#272

Splittings

Fitting	C₂×C₄×H(3,𝔽₃) ⋊ C₂
Sylow	H(3,𝔽₃) ⋊ C₂×D₄

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
hypercenter	C₂×C₃×C₄	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₂×D₄
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄

297 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11720324, #11720325, #11720531, …
maximal normal subgroup of	#11720324, #11720325, #11720531, …
maximal quotient of	#11721715, C₄×C₄⋊#272, #11721720, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 386);