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C₂×3₋¹⁺²⋊C₉ #92233

order	486
nilpotency class	3
isoclinism	isoclinic class of AΣL(1,𝔽₂₇)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₉ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₂×3₋¹⁺²⋊C₉
unambiguous semidirect product	(C₂×3₋¹⁺²)⋊C₉

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	H(3,𝔽₃)
commutator	C₃²	C₂×C₃×C₉
Frattini, cosocle	C₃³	C₂×C₃²
prime radical	C₂×C₃³	C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃³ ◁ G

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 2 generators on 20 points:

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

derived subgroup of	(3₋¹⁺²⋊C₉)⋊Q₈, C₄⋊((3₋¹⁺²⋊C₉)⋊C₂), #14516183
Fitting subgroup of	(3₋¹⁺²⋊C₉)⋊C₄, C₂×(3₋¹⁺²⋊C₉)⋊C₂
maximal subgroup of	(3₋¹⁺²⋊C₉)⋊C₄, C₄×3₋¹⁺²⋊C₉, C₂×(3₋¹⁺²⋊C₉)⋊C₂, …
maximal normal subgroup of	(3₋¹⁺²⋊C₉)⋊C₄, C₄×3₋¹⁺²⋊C₉, C₂×(3₋¹⁺²⋊C₉)⋊C₂, …
maximal quotient of	C₄×3₋¹⁺²⋊C₉, C₂²×3₋¹⁺²⋊C₉, C₂×3₋¹⁺²⋊C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 78);