C₉×C₂₇⋊C₃ #10624680

order	729
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃×C₉² · Schur multiplier (H₂) = C₃×C₉

Aliases

direct product

C₉×C₂₇⋊C₃

Subgroups

	subgroup	G/H
center	C₉²	C₃²
commutator	C₃	C₃×C₉²
Frattini, cosocle	C₃×C₉	C₃³
socle	C₃²	C₃²×C₉
prime radical	C₃³	C₃×C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉² ◁ G
LECS	G ▷ C₃×C₉ ▷ C₃ ▷ C₁

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 243

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2,3,4,5,6,7,8,9);(10,11,14,12,16,22,18,25,30,13,17,21,19,26,29,27,33,35,20,15,23,28,24,31,34,32,36);(11,15,17)(14,21,23)(16,24,26)(22,29,31)(25,32,33)(30,35,36)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	D₉×C₂₇⋊C₃, C₉×D₂₇⋊C₃, #13090865
maximal subgroup of	C₂×C₉×C₂₇⋊C₃, D₉×C₂₇⋊C₃, C₉×D₂₇⋊C₃, …
maximal normal subgroup of	C₂×C₉×C₂₇⋊C₃, D₉×C₂₇⋊C₃, C₉×D₂₇⋊C₃, …
maximal quotient of	C₂×C₉×C₂₇⋊C₃, C₂₇×C₂₇⋊C₃, #14546554, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(729, 240);

C9×C27⋊C3 #10624680