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H(3,𝔽₃)×C₃₂ #11720104

order	864
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₃₂ · Schur multiplier (H₂) = C₃²

Aliases

direct product	H(3,𝔽₃)×C₃₂
unambiguous semidirect product	(C₃²×C₃₂)⋊C₃

Subgroups

	subgroup	G/H
center	C₃×C₃₂	C₃²
commutator	C₃	C₃²×C₃₂
Frattini, cosocle	C₃×C₁₆	C₂×C₃²
socle	C₂×C₃	C₃²×C₁₆
prime radical	C₂×H(3,𝔽₃)	C₁₆
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₃₂ ◁ G

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 41
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 2 generators on 41 points:

(1,2,4)(3,6,8)(5,9,7)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41);(2,5,6)(4,8,9)

Conjugacy classes & character table

352 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	(C₃⋊(C₃⋊C₆₄))⋊C₃, #13134122, C₃₂×(C₃⋊D₃)⋊C₃, …
maximal subgroup of	(C₃⋊(C₃⋊C₆₄))⋊C₃, #13134122, H(3,𝔽₃)×C₆₄, …
maximal normal subgroup of	(C₃⋊(C₃⋊C₆₄))⋊C₃, #13134122, H(3,𝔽₃)×C₆₄, …
maximal quotient of	H(3,𝔽₃)×C₆₄, C₂×H(3,𝔽₃)×C₃₂, C₃×H(3,𝔽₃)×C₃₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 71);