#72243485

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃⁵

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#14555275 ⋊ C₂
derived	C₃²×C₉ ⋊ C₂×C₃³

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²×C₉	C₂×C₃³
Frattini, socle	C₃²	C₃³×C₃⋊D₃
Fitting	#14555275	C₂
LCS	G ▷ C₃²×C₉
UCS	C₁

3952 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 3952 conjugacy classes of subgroups — too many to draw. Only its 227 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) = 81

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

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

Polycyclic presentation:

2,3^7|f^3=h,e^a=e^2,e^c=eh,e^d=eg,f^a=f^2h^2,f^b=fh,f^c=fh,f^d=fh^2,g^a=g^2,h^a=h^2

Conjugacy classes & character table

198 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72243486

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