C₃²×#6875 #14554148

order	2187
nilpotency class	3
isoclinism	isoclinic class of #6875
automorphism	?
homology	Abelianization (H₁) = C₃³×C₉ · Schur multiplier (H₂) = C₃⁵

Aliases

direct product

C₃²×#6875

Subgroups

	subgroup	G/H
center, socle	C₃³	C₉⋊C₉
commutator	C₉	C₃³×C₉
Frattini, cosocle	C₃×C₉	C₃⁴
prime radical	C₃⁴	3₋¹⁺²
LCS	G ▷ C₉ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³ ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃×C₉ ▷ C₃ ▷ C₁

872 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);(8,11,17,13,20,29,24,32,19)(9,14,25)(12,21,18,22,33,30,23,28,31)(15,27,26);(7,8,12,9,11,18,15,19,21,10,13,23,14,20,31,26,17,28,16,24,22,25,32,30,27,29,33)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₃⋊D₃×#6875, C₃²×AGL(1,ℤ/27), #72243558, …
maximal subgroup of	C₃⋊D₃×#6875, C₃²×AGL(1,ℤ/27), #72243558, …
maximal normal subgroup of	C₃⋊D₃×#6875, C₃²×AGL(1,ℤ/27), #72243558, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2187, 8007);

C32×#6875 #14554148