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#72227657

order1536
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, Frattini, socleC23C2×#1020
commutator, prime radical, cosocleC2×#543⋊C3C22
FittingC2×#3458D3
hypercenterC22×C4#1020
hypocenter#543⋊C3D4
LCSG ▷ C2×#543⋊C3 ▷ #543⋊C3
UCSC1 ◁ C23 ◁ C22×C4

1228 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 1228 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 128

A minimal faithful permutation representation — 2 generators on 36 points:

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

Polycyclic presentation:

2^3,3,2^6|a^2=c,b^2=j,e^2=i,f^2=i,b^a=bci,d^a=d^2,e^a=fi,e^d=ef,f^a=ei,f^d=e,f^e=fi,g^a=gh,g^d=h,g^f=gj,h^d=gh,h^e=hj,h^f=hj

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227666

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.6*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.5*F.9);;
gap> SetConjugate(coll, 6, 4, F.5);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);