Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | #6629044 ⋊ C3 |
1659 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 1659 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(1,3,14,38,7,18,4,15)(2,9,19,35,8,28,10,13)(5,17,51,55,27,44,20,42,26,16,40,63,24,48,52,41,6,47,39,57,23,45,21,64)(11,30,59,43,25,56,31,37,34,29,50,61,22,54,60,36,12,58,49,46,33,53,32,62);(1,23,7,26)(2,34,8,33)(3,44,18,47)(4,51,14,52)(5,27,24,6)(9,58,28,56)(10,60,19,59)(11,12,22,25)(13,62,35,61)(15,63,38,64)(16,48,45,17)(20,40,39,21)(29,30,53,54)(31,32,49,50)(36,37,43,46)(41,57,55,42);(1,2)(3,13)(4,19)(5,22)(6,25)(7,8)(9,15)(10,14)(11,24)(12,27)(16,43)(17,46)(18,35)(20,31)(21,32)(23,34)(26,33)(28,38)(29,55)(30,57)(36,45)(37,48)(39,49)(40,50)(41,53)(42,54)(44,62)(47,61)(51,59)(52,60)(56,64)(58,63)Polycyclic presentation:
2^3,3,2^6|c^2=ej,e^2=j,f^2=j,g^2=j,b^a=bj,c^a=ce,e^a=ej,f^a=fi,f^d=fg,g^a=gh,g^d=f,g^f=gj,h^d=i,h^f=hj,i^d=hij,i^g=ij68 conjugacy classes — view the full character table.
| maximal subgroup of | #72341798, #72341841, #72341888, … |
| maximal normal subgroup of | #72341798, #72341841, #72341888, … |
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, 3, F.5*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.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.6);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);