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C₈×#272 #88526

order	432
isoclinism	isoclinic class of #272
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

Splittings

Fitting	C₈×H(3,𝔽₃) ⋊ C₂
derived, Sylow	H(3,𝔽₃) ⋊ C₂×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₈	C₃⋊D₃
commutator, hypocenter	H(3,𝔽₃)	C₂×C₈
Frattini	C₃×C₄	C₂×C₃⋊D₃
Fitting	C₈×H(3,𝔽₃)	C₂
socle	C₂×C₃	C₄×C₃⋊D₃
prime radical	C₂×#272	C₄
cosocle	C₄×H(3,𝔽₃)	C₂²
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₈

121 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 3 generators on 17 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11720251, #11720252, C₁₆×#272, …
maximal normal subgroup of	#11720251, #11720252, C₁₆×#272, …
maximal quotient of	C₁₆×#272, #11720511, C₈×#1084, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 173);