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#72267752

order1536
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#104709 ⋊ C3
derived#17072 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC22#1655⋊C3
commutator, cosocle, hypocenter#17072C2×C3
FrattiniC22×C42C2×A4
Fitting#104709C3
socleC24#120⋊C3
LCSG ▷ #17072
UCSC1 ◁ C22

453 conjugacy classes of subgroups; only the 19 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 19 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) = 32
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2,3,2^8|c^2=g,d^2=ghi,e^2=i,f^2=h,b^a=bi,c^a=cf,c^b=cd,d^a=de,d^b=ci,d^c=dgh,e^a=ei,e^b=f,e^c=egj,e^d=ei,f^a=fh,f^b=efgj,f^c=fh,f^d=fgj,g^b=gi,h^b=hi,i^b=h,j^b=hij

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72267755, #72267774

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