#72243278

order	13122
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃³

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

3₋¹⁺²×#6900 ⋊ C₂

Subgroups

	subgroup	G/H
center	C₃	C₃×#13091521
commutator, cosocle	C₃×C₉²	C₂×C₃³
Frattini, socle	C₃³	C₃³×C₃⋊D₃
Fitting	3₋¹⁺²×#6900	C₂
prime radical	C₃×#13091521	C₃
hypercenter	C₉	#13091521
hypocenter	C₉²	C₂×C₃×3₋¹⁺²
LCS	G ▷ C₃×C₉² ▷ C₉²
UCS	C₁ ◁ C₃ ◁ C₉

6046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 162

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

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

Polycyclic presentation:

2,3^8|d^3=e,f^3=h,g^3=i,d^a=dh^2,d^c=de,f^a=f^2h^2,f^c=fh,g^a=g^2i^2,g^b=gi,h^a=h^2,i^a=i^2

Conjugacy classes & character table

330 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/9)×#92265, #72243365, #72243381
maximal normal subgroup of	AGL(1,ℤ/9)×#92265, #72243365, #72243381

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