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#72243792

order8748
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#14554266 ⋊ C22
derived#10624732 ⋊ C22×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#10624732C22×C3
FrattiniC33C3×D3×C3⋊D3
Fitting#14554266C22
socleC32C3×((C3⋊D3)⋊C3)⋊D3
LCSG ▷ #10624732
UCSC1

2199 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 2199 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 54

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72243793

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