#72243754

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃⁵

Aliases

No aliases for this group.

Splittings

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

Subgroups

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

4180 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 4180 conjugacy classes of subgroups — too many to draw. Only its 231 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 — 4 generators on 36 points:

(2,4)(3,6)(5,7)(8,9)(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);(2,5,9)(4,7,8)(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);(1,2,4)(3,5,8)(6,9,7)

Polycyclic presentation:

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

Conjugacy classes & character table

171 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243994
maximal normal subgroup of	#72243994
maximal quotient of	#72243764

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, 5, F.8);;
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);;
gap> SetConjugate(coll, 6, 2, 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);