C₃²×C₂₇⋊C₃ #10624916

order	729
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃³×C₉ · Schur multiplier (H₂) = C₃⁵

Aliases

direct product	C₃²×C₂₇⋊C₃
unambiguous semidirect product	C₂₇⋊C₃³

Subgroups

	subgroup	G/H
center	C₃²×C₉	C₃²
commutator	C₃	C₃³×C₉
Frattini, cosocle	C₉	C₃⁴
socle	C₃³	C₃×C₉
prime radical	C₃⁴	C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₉ ▷ C₃ ▷ C₁

472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 243

A minimal faithful permutation representation — 4 generators on 33 points:

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

Conjugacy classes & character table

297 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	C₃×D₃×C₂₇⋊C₃, C₃⋊D₃×C₂₇⋊C₃, C₃²×D₂₇⋊C₃, …
maximal subgroup of	C₂×C₃²×C₂₇⋊C₃, C₃×D₃×C₂₇⋊C₃, C₃⋊D₃×C₂₇⋊C₃, …
maximal normal subgroup of	C₂×C₃²×C₂₇⋊C₃, C₃×D₃×C₂₇⋊C₃, C₃⋊D₃×C₂₇⋊C₃, …
maximal quotient of	C₂×C₃²×C₂₇⋊C₃, C₃²×#6864, C₃×C₉×C₂₇⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(729, 476);

C32×C27⋊C3 #10624916