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C₄×C₈×H(3,𝔽₃) #11720591

order	864
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄×C₈ · Schur multiplier (H₂) = C₃²×C₄

Aliases

direct product	C₄×C₈×H(3,𝔽₃)
unambiguous semidirect product	(C₃²×C₄×C₈)⋊C₃

Subgroups

	subgroup	G/H
center	C₃×C₄×C₈	C₃²
commutator	C₃	C₃²×C₄×C₈
Frattini, cosocle	C₂×C₃×C₄	C₂²×C₃²
socle	C₂²×C₃	C₂×C₃²×C₄
prime radical	C₂²×H(3,𝔽₃)	C₂×C₄
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₈ ◁ G

242 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 2 generators on 21 points:

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

Conjugacy classes & character table

352 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₈×(C₃⋊(C₃⋊C₈))⋊C₃, #10589320⋊C₃, #10589321⋊C₃, …
maximal subgroup of	C₈×(C₃⋊(C₃⋊C₈))⋊C₃, #10589320⋊C₃, #10589321⋊C₃, …
maximal normal subgroup of	C₈×(C₃⋊(C₃⋊C₈))⋊C₃, #10589320⋊C₃, #10589321⋊C₃, …
maximal quotient of	C₈²×H(3,𝔽₃), H(3,𝔽₃)×#322, H(3,𝔽₃)×#336, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 558);