Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
798 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 798 conjugacy classes of subgroups — too many to draw. Only its 29 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:
(1,30,56,48)(2,40,59,43)(3,20,64,32)(4,47,27,18)(5,45,31,26)(6,52,21,16)(7,19,23,63)(8,17,54,46)(9,53,61,29)(10,49,41,38)(11,58,55,14)(12,35,36,62)(13,33,60,51)(15,28,50,24)(22,42,39,34)(25,57,37,44);(1,2,4,10,8,13,5,22)(3,14,15,35,19,44,16,9)(6,29,56,53,32,41,23,37,31,25,21,59,28,12,54,36,7,39,20,55,27,11,24,60)(17,51,64,33,26,62,47,38,52,49,46,58,30,43,63,40,18,61,45,34,50,42,48,57)Polycyclic presentation:
2^4,3,2^5|a^2=c,b^2=d,c^2=j,d^2=j,f^2=j,g^2=j,b^a=bd,d^a=dj,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=ij50 conjugacy classes — view the full character table.
| maximal subgroup of | #72341825 |
| maximal normal subgroup of | #72341825 |
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.3);;
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.4);;
gap> SetConjugate(coll, 4, 1, F.4*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);