#72243764

order	13122
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃⁵

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#72243632 ⋊ C₂
derived	C₃²×C₉² ⋊ C₂×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²×C₉²	C₂×C₃²
Frattini, socle	C₃³	C₃²×C₃⋊(C₃⋊D₃)
Fitting	#72243632	C₂
LCS	G ▷ C₃²×C₉²
UCS	C₁

8554 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 729

A minimal faithful permutation representation — 4 generators on 27 points:

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

Polycyclic presentation:

2,3^8|e^3=h,f^3=i,d^a=d^2,d^b=dg,e^a=e^2h^2,e^b=eh,e^c=eh,f^a=f^2i^2,f^b=fh^2,f^c=fi,g^a=g^2,h^a=h^2,i^a=i^2

Conjugacy classes & character table

246 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243996, #72254071
maximal normal subgroup of	#72243996, #72254071

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