#72243996

order	26244
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃³

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#72243632 ⋊ C₂²
derived	C₃²×C₉² ⋊ C₂²×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²×C₉²	C₂²×C₃²
Frattini, socle	C₃³	C₃²×D₃×C₃⋊D₃
Fitting	#72243632	C₂²
LCS	G ▷ C₃²×C₉²
UCS	C₁

6936 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 486

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,4)(3,5,8)(6,9,7)(11,17,18,13,14,15)(12,16)(20,27,26,21,24,25)(22,23);(2,7,9,4,5,8)(3,6)(10,14,13,12,18,15,16,11,17)(19,27,26,23,25,24,22,21,20);(19,24,21,22,26,25,23,20,27)

Polycyclic presentation:

2^2,3^8|f^3=i,g^3=j,e^a=e^2,e^c=eh,f^b=f^2i^2,f^c=fi,f^d=fi,g^b=g^2j^2,g^c=gi^2,g^d=gj,h^a=h^2,i^b=i^2,j^b=j^2

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72269259
maximal normal subgroup of	#72269259

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", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6^2*F.9^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7^2*F.10^2);;
gap> SetConjugate(coll, 7, 3, F.7*F.9^2);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> SetConjugate(coll, 10, 2, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);