Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | #6372742 ⋊ C3 |
1659 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 1659 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 64 points:
(1,4,8,5)(2,10,13,20)(3,15,19,16)(6,51,30,23,26,22,27,52,7,21,29,24)(9,31,14,38)(11,57,28,49,36,35,25,58,12,34,37,50)(17,61,48,43,45,42,46,62,18,41,47,44)(32,63,40,59,55,54,39,64,33,53,56,60);(1,47,4,62,8,45,5,61)(2,56,20,63,13,55,10,64)(3,51,15,29,19,52,16,26)(6,18,21,42,27,48,23,44)(7,46,22,43,30,17,24,41)(9,58,38,37,14,57,31,36)(11,33,49,60,25,40,34,54)(12,39,50,53,28,32,35,59);(1,2,5,10,8,13,4,20)(3,14,16,38,19,9,15,31)(6,25,23,49,27,11,21,34)(7,28,24,50,30,12,22,35)(17,32,43,53,46,39,41,59)(18,33,44,54,48,40,42,60)(26,37,52,58,29,36,51,57)(45,55,62,63,47,56,61,64)Polycyclic presentation:
2^3,3,2^6|a^2=e,b^2=e,c^2=j,e^2=j,f^2=j,g^2=j,b^a=bj,c^a=cj,f^a=fhi,f^d=g,g^a=gh,g^d=fgj,g^f=gj,h^d=ij,h^f=hj,i^d=hij,i^f=ij,i^g=ij92 conjugacy classes — view the full character table.
| maximal subgroup of | #72341841, #72342728, #72342787, … |
| maximal normal subgroup of | #72341841, #72342728, #72342787, … |
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.5);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.10);;
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, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.9*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, 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);