C₉×#6875 #14550273

order	2187
nilpotency class	3
isoclinism	isoclinic class of #6875
automorphism	?
homology	Abelianization (H₁) = C₃×C₉² · Schur multiplier (H₂) = C₃×C₉

Aliases

direct product

C₉×#6875

Subgroups

	subgroup	G/H
center	C₃×C₉	C₉⋊C₉
commutator	C₉	C₃×C₉²
Frattini, cosocle	C₃²×C₉	C₃³
socle	C₃²	C₃×C₉⋊C₉
prime radical	C₃³	C₃×3₋¹⁺²
LCS	G ▷ C₉ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉ ◁ C₃×C₉² ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

288 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);(11,14,20,16,23,32,27,35,22)(12,17,28)(15,24,21,25,36,33,26,31,34)(18,30,29);(10,11,15,12,14,21,18,22,24,13,16,26,17,23,34,29,20,31,19,27,25,28,35,33,30,32,36)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₉×AGL(1,ℤ/27), C₂×C₉×#6875
maximal normal subgroup of	C₉×AGL(1,ℤ/27), C₂×C₉×#6875
maximal quotient of	C₂×C₉×#6875

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2187, 4132);

C9×#6875 #14550273