Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting, Sylow | #495678 ⋊ C3 |
388 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,3,9,4,10,13,28,7,8,16,24,19,25,37,46)(5,33,15,29,43,32,35,63,22,26,41,48,42,56,59,61,21,12,14,53,18,31,58,52,20,34,40,47,45,50,57,64,6,11,39,30,17,55,36,51,23,27,38,54,44,49,60,62);(1,23)(2,33)(3,41)(4,45)(5,22,20,6)(7,21)(8,34)(9,53)(10,55)(11,12,26,27)(13,60)(14,40,38,15)(16,39)(17,44,42,18)(19,43)(24,54)(25,56)(28,63)(29,30,47,48)(31,32,49,50)(35,59,57,36)(37,58)(46,64)(51,52,61,62)Polycyclic presentation:
2^2,3,2^7|b^2=d,d^2=e,e^2=j,f^2=j,g^2=j,b^a=bj,f^a=fhi,f^c=g,g^a=gh,g^c=fgj,g^f=gj,h^c=ij,h^f=hj,i^c=hij,i^f=ij,i^g=ij92 conjugacy classes — view the full character table.
| maximal subgroup of | #72342027 |
| maximal normal subgroup of | #72342027 |
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, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.9*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);