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C₄×#1084 #88539

order	432
isoclinism	isoclinic class of #272
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

Splittings

H(3,𝔽₃) ⋊ C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄	C₃⋊D₃
commutator, hypocenter	H(3,𝔽₃)	C₄²
Frattini, socle	C₂²×C₃	C₂×C₃⋊D₃
Fitting	C₂×C₄×H(3,𝔽₃)	C₂
prime radical, cosocle	C₂²×H(3,𝔽₃)	C₂²
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃×C₄

165 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 144

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

(2,4)(5,9)(6,8)(14,15,16,17);(2,5,6)(4,8,9)(10,11,12,13);(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	#11720302, #11720303, #11720304, …
maximal normal subgroup of	#11720302, #11720303, #11720304, …
maximal quotient of	C₄×#6180, #11720520, C₈×#1084, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 186);