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H(3,𝔽₃)×#108 #11720627

order	864
nilpotency class	3
isoclinism	isoclinic class of D₈×H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₄ · Schur multiplier (H₂) = C₃²

Aliases

direct product	H(3,𝔽₃)×#108
unambiguous semidirect product	(C₃²×#108)⋊C₃

Subgroups

	subgroup	G/H
center	C₃×C₄	C₃²×D₄
commutator	C₃×C₄	C₂×C₃²×C₄
Frattini, cosocle	C₂×C₃×C₄	C₂²×C₃²
socle	C₂×C₃	C₃²×C₄⋊C₄
prime radical	C₂²×H(3,𝔽₃)	Q₈
LCS	G ▷ C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₄×H(3,𝔽₃) ◁ G

165 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 25 points:

(1,2,4)(3,6,8)(5,9,7)(10,11,14,17,15,18,13,23)(12,19,21,24,22,25,20,16);(2,5,6)(4,8,9)(10,12,13,20,15,22,14,21)(11,16,17,19,18,24,23,25)

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#10589359⋊C₃, #10589360⋊C₃, #10589361⋊C₃, …
maximal subgroup of	#10589359⋊C₃, #10589360⋊C₃, #10589361⋊C₃, …
maximal normal subgroup of	#10589359⋊C₃, #10589360⋊C₃, #10589361⋊C₃, …
maximal quotient of	H(3,𝔽₃)×#334, H(3,𝔽₃)×#335, H(3,𝔽₃)×#341, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 594);