#72228249

order	6561
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂₇² · Schur multiplier (H₂) = C₃×C₉²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₃²×C₉	C₉²
commutator	C₉	C₂₇²
Frattini, cosocle	C₉³	C₃²
socle, prime radical	C₃³	#6855
LCS	G ▷ C₉ ▷ C₁
UCS	C₁ ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₉³ ▷ C₃³ ▷ C₁

438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 135
transitive degree	μ_t(G) = 6561

A minimal faithful permutation representation — 2 generators on 135 points:

(1,2,5,3,6,11,8,13,18,4,7,12,9,14,19,16,21,24,10,15,20,17,22,25,23,26,27)(55,56,60)(57,64,83,66,85,111,68,86,112)(58,63,74)(59,62,75)(61,76,93,78,104,124,79,105,125)(65,88,118,89,119,103,91,122,130)(67,87,98,92,116,99,94,117,72)(69,82,101)(70,81,102)(71,80,73)(77,106,123,107,131,90,108,132,120)(84,113,134,114,97,121,115,127,135)(95,110,129)(96,109,100)(126,133,128);(28,29,32,30,33,38,35,40,45,31,34,39,36,41,46,43,48,51,37,42,47,44,49,52,50,53,54)(55,57,65)(56,61,77,62,79,107,80,78,108)(58,67,90)(59,68,89)(60,72,97,73,98,113,75,99,127)(63,64,84,81,86,114,109,85,115)(66,91,71)(69,93,121)(70,94,120)(74,76,103,100,104,118,102,105,130)(82,87,88,110,117,119,133,116,122)(83,106,128,111,132,129,112,131,101)(92,123,96)(95,125,135)(124,134,126)

Polycyclic presentation:

3^8|a^3=c,b^3=d,c^3=f,d^3=g,e^3=h^2,b^a=be,c^b=ch,d^a=dh^2

Conjugacy classes & character table

945 conjugacy classes — view the full character table.

Relationships

maximal quotient of

H(3,ℤ/27)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h");;
gap> coll := SingleCollector(F, [3, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8^2);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);