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H(3,𝔽₃)×#100 #11720603

order	864
nilpotency class	3
isoclinism	isoclinic class of H(3,𝔽₃)×#99
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₄ · Schur multiplier (H₂) = C₂×C₃²

Aliases

direct product	H(3,𝔽₃)×#100
unambiguous semidirect product	(C₃²×#100)⋊C₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₃²×C₂²⋊C₄
commutator	C₂²×C₃	C₂×C₃²×C₄
Frattini, cosocle	C₂×C₃×C₄	C₂²×C₃²
prime radical	C₂×D₄×H(3,𝔽₃)	C₂
LCS	G ▷ C₂²×C₃ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₄×H(3,𝔽₃) ◁ G

253 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 17 points:

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

Conjugacy classes & character table

121 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#10589342⋊C₃, #10589341⋊C₃, #13135780, …
maximal subgroup of	#10589342⋊C₃, #10589341⋊C₃, #13135780, …
maximal normal subgroup of	#10589342⋊C₃, #10589341⋊C₃, #13135780, …
maximal quotient of	H(3,𝔽₃)×#323, H(3,𝔽₃)×#329, H(3,𝔽₃)×#331, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 570);