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#72268136

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, Frattini, socleC22C22×#1020
commutator, cosocle, hypocenter#543⋊C3C23
Fitting#62844D3
prime radical#62844⋊C3C2
hypercenterC22⋊C4#1020
LCSG ▷ #543⋊C3
UCSC1 ◁ C22 ◁ C22⋊C4

1871 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 1871 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 48
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,15,11,6,9,8,5,16,4,2,12,14)(3,7,10,13)(18,24)(19,27)(20,33,43)(21,31,32)(25,26,45,37,38,40)(28,29,44,41,42,30)(34,47,48)(35,39,46);(3,9,10,12)(7,15,13,16)(17,30,33,41)(18,39,38,46)(19,43,42,20)(21,28,22,44)(23,45,47,25)(24,48,26,34)(27,31,29,32)(35,37,36,40);(2,6)(7,13)(8,14)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,36)(28,34)(29,35)(30,39)(31,40)(32,37)(33,38)(41,46)(42,47)(43,45)(44,48)

Polycyclic presentation:

2^3,3,2^6|b^2=j,c^2=ij,g^2=ij,h^2=ij,c^a=ci,c^b=cj,d^b=d^2,e^b=ef,e^d=f,f^d=ef,g^b=gh,g^d=h,g^f=gi,h^b=hij,h^d=ghij,h^e=hi,h^g=hij

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72268139

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