#72243593

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72243517 ⋊ C₂²

Subgroups

	subgroup	G/H
center	C₃	C₃×AGL(1,ℤ/9)²
commutator, cosocle	C₃×C₉²	C₂²×C₃³
Frattini, socle	C₃³	C₃³×D₃²
Fitting	#72243517	C₂²
prime radical	C₃²×#92265⋊C₂	C₃
hypercenter	C₃²	AGL(1,ℤ/9)²
hypocenter	C₉²	C₂²×C₃×3₋¹⁺²
LCS	G ▷ C₃×C₉² ▷ C₉²
UCS	C₁ ◁ C₃ ◁ C₃²

5354 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:

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

Polycyclic presentation:

2^2,3^8|c^3=f,g^3=i,h^3=j,d^c=df,g^a=g^2i^2,g^b=g^2i^2,g^c=gi^2,g^e=gi^2,h^a=g^2h,h^b=h^2j^2,h^c=hi,h^e=hj^2,i^a=i^2,i^b=i^2,j^a=i^2j,j^b=j^2

Conjugacy classes & character table

372 conjugacy classes — view the full character table.

Relationships

holomorph of

D₉⋊C₉

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