#72269259

order	52488
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃²

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#72243632 ⋊ C₂³
derived	C₃²×C₉² ⋊ C₂³×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²×C₉²	C₂³×C₃²
Frattini, socle	C₃³	C₃²×D₃³
Fitting	#72243632	C₂³
LCS	G ▷ C₃²×C₉²
UCS	C₁

8928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 108

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

Polycyclic presentation:

2^3,3^8|g^3=j,h^3=k,f^b=f^2,f^d=fi,g^a=g^2j^2,g^c=g^2j^2,g^d=gj,g^e=gj,h^a=g^2h,h^c=h^2k^2,h^d=hj^2,h^e=hk,i^b=i^2,j^a=j^2,j^c=j^2,k^a=j^2k,k^c=k^2

Conjugacy classes & character table

336 conjugacy classes — view the full character table.

Relationships

holomorph of

C₃×AGL(1,ℤ/9)

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.10^2);;
gap> SetConjugate(coll, 7, 3, F.7^2*F.10^2);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 3, F.8^2*F.11^2);;
gap> SetConjugate(coll, 8, 4, F.8*F.10^2);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> SetConjugate(coll, 11, 1, F.10^2*F.11);;
gap> SetConjugate(coll, 11, 3, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);