Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #20612 ⋊ D3 |
1408 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 49 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:
(1,23,27,5,28,52,8,21,31,4,6,51)(2,49,39,20,25,60,13,36,41,10,11,58)(3,45,47,16,30,62,19,43,48,15,17,61)(7,22,32,24)(9,59,55,42,40,64,14,53,56,33,34,63)(12,37,29,50)(18,44,26,46)(35,54,38,57);(1,3,5,16,8,19,4,15)(2,9,10,33,13,14,20,42)(6,26,23,44,28,18,21,46)(7,30,24,43,32,17,22,45)(11,38,36,57,25,35,49,54)(12,40,37,59,29,34,50,53)(27,48,52,61,31,47,51,62)(39,56,58,64,41,55,60,63);(1,2,5,10,8,13,4,20)(3,14,16,42,19,9,15,33)(6,25,23,49,28,11,21,36)(7,29,24,50,32,12,22,37)(17,34,45,53,30,40,43,59)(18,35,46,54,26,38,44,57)(27,41,52,60,31,39,51,58)(47,55,62,63,48,56,61,64)Polycyclic presentation:
2^4,3,2^5|a^2=d,b^2=d,c^2=j,d^2=j,f^2=j,g^2=j,b^a=bj,c^a=cj,e^b=e^2,f^a=fhij,f^b=gj,f^e=g,g^a=ghj,g^b=fj,g^e=fgj,g^f=gj,h^b=hi,h^e=i,h^f=hj,i^e=hi,i^f=ij,i^g=ij62 conjugacy classes — view the full character table.
| maximal subgroup of | #72341861, #72342758, #72342787, … |
| maximal normal subgroup of | #72341861, #72342758, #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, 2, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, 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, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.7*F.10);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
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);