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C₂²×#1084 #88751

order	432
isoclinism	isoclinic class of #272
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

H(3,𝔽₃) ⋊ C₂²×C₄

Subgroups

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

297 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:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11720334, #11720574, C₂²⋊#6188, …
maximal normal subgroup of	#11720334, #11720574, C₂²⋊#6188, …
maximal quotient of	C₂²×#6180, C₂×#88538, C₂×C₄×#1084, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 398);