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C₂×#6199 #88674

order	432
isoclinism	isoclinic class of ((C₃⋊D₃)⋊C₃)⋊C₂
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

H(3,𝔽₃) ⋊ C₂²×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	((C₃⋊D₃)⋊C₃)⋊C₂
commutator, hypocenter	H(3,𝔽₃)	C₂²×C₄
Frattini	C₂×C₃	C₂×D₃²
Fitting	C₂²×H(3,𝔽₃)	C₂²
socle	C₂²×C₃	D₃²
prime radical	C₂²×#272	C₂
cosocle	C₂×H(3,𝔽₃)	C₂³
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²

221 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:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11720707, #11721076, #11721102, …
maximal normal subgroup of	#11720707, #11721076, #11721102, …
maximal quotient of	C₂×#88434, #11721064, #11721065, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 321);