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C₄²×H(3,𝔽₃) #88554

order	432
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄² · Schur multiplier (H₂) = C₃²×C₄

Aliases

C₄²×H(3,𝔽₃)

Subgroups

	subgroup	G/H
center	C₃×C₄²	C₃²
commutator	C₃	C₃²×C₄²
Frattini, socle, cosocle	C₂²×C₃	C₂²×C₃²
prime radical	C₂²×H(3,𝔽₃)	C₂²
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₄² ◁ G

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 — 2 generators on 17 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₄×(C₃⋊(C₃⋊C₈))⋊C₃, #63630⋊C₃, C₄⋊((C₃⋊(C₃⋊C₈))⋊C₃), …
maximal subgroup of	C₄×(C₃⋊(C₃⋊C₈))⋊C₃, #63630⋊C₃, C₄⋊((C₃⋊(C₃⋊C₈))⋊C₃), …
maximal normal subgroup of	C₄×(C₃⋊(C₃⋊C₈))⋊C₃, #63630⋊C₃, C₄⋊((C₃⋊(C₃⋊C₈))⋊C₃), …
maximal quotient of	H(3,𝔽₃)×#95, C₄×C₈×H(3,𝔽₃), H(3,𝔽₃)×#97, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 201);