#72243994

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃³

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#14555275 ⋊ C₂²
derived	C₃³×C₉ ⋊ C₂²×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃³×C₉	C₂²×C₃²
Frattini, socle	C₃²	C₃²×D₃×C₃⋊D₃
Fitting	#14555275	C₂²
LCS	G ▷ C₃³×C₉
UCS	C₁

3331 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 3331 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2,4)(3,5,8)(6,9,7)(11,22,24,14,15,16)(12,18)(13,21)(17,33,30,27,25,36)(19,29)(20,31)(23,34,26,28,32,35);(2,7,9,4,5,8)(3,6)(10,12,31)(11,25,34)(13,20,18)(14,32,27)(15,30,35)(16,23,36)(17,28,24)(19,29,21)(22,26,33);(10,15,14,13,24,16,21,11,22)(12,25,23,20,30,26,19,17,32)(18,34,27,29,35,36,31,28,33)

Polycyclic presentation:

2^2,3^7|f^3=i,e^a=e^2,e^c=eh,f^b=f^2i^2,f^c=fi,f^d=fi,g^b=g^2,g^c=gi^2,h^a=h^2,i^b=i^2

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72243996

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