Finite Group DatabasebetaFinite Group Databasebeta| order | 10816 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C132 ⋊ #428 |
512 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 512 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 104 |
Polycyclic presentation:
2^6,13^2|a^2=e,b^2=d,e^2=f,b^a=bef,c^a=cf,e^b=ef,g^a=g^12,g^b=g^8,g^d=g^12,h^a=g^6h,h^b=h^8,h^d=h^12226 conjugacy classes — view the full character table.
| maximal subgroup of | #72259289, #72259548 |
| maximal normal subgroup of | #72259289, #72259548 |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 13, 13]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^12);;
gap> SetConjugate(coll, 7, 2, F.7^8);;
gap> SetConjugate(coll, 7, 4, F.7^12);;
gap> SetConjugate(coll, 8, 1, F.7^6*F.8);;
gap> SetConjugate(coll, 8, 2, F.8^8);;
gap> SetConjugate(coll, 8, 4, F.8^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);