C₉×#6887 #14552010

order	2187
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₉² · Schur multiplier (H₂) = C₃³×C₉²

Aliases

direct product

C₉×#6887

Subgroups

	subgroup	G/H
center	C₃×C₉²	C₃²
commutator	C₃	C₃²×C₉²
Frattini, socle, cosocle	C₃³	C₃⁴
prime radical	C₃⁴	C₃³
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉² ◁ G
LECS	G ▷ C₃³ ▷ C₁

1583 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 729

A minimal faithful permutation representation — 4 generators on 45 points:

(1,2,3,4,5,6,7,8,9);(10,11,13,12,14,16,15,17,18)(19,20,23)(21,25,31)(22,26,30)(24,32,29)(27,34,39)(28,35,38)(33,40,37)(36,42,44)(41,45,43);(20,24,26)(23,30,32)(25,33,35)(31,38,40)(34,41,42)(39,44,45);(19,21,27,22,28,36,29,37,43)(20,25,34,26,35,42,24,33,41)(23,31,39,30,38,44,32,40,45)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72270942, C₉×H(3,ℤ/9)
maximal normal subgroup of	#72270942, C₉×H(3,ℤ/9)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2187, 5869);

C9×#6887 #14552010