Finite Group DatabasebetaFinite Group Databasebeta| order | 4992 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C13×#100 ⋊ C3×C4 |
| Hall | #13130791 ⋊ C3 |
| Hall | C13⋊C3 ⋊ #1856 |
| Sylow | C13 ⋊ C3×#1856 |
648 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 648 conjugacy classes of subgroups — too many to draw. Only its 124 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:
2^3,3,2^4,13|a^2=e,b^2=e,g^2=h,b^a=bg,c^a=ch,c^b=cf,f^a=fh,g^a=gh,g^b=gh,g^c=gh,i^a=i^8,i^b=i^8,i^d=i^3,i^e=i^12107 conjugacy classes — view the full character table.
| maximal subgroup of | #453×AGL(1,𝔽13) |
| maximal normal subgroup of | #453×AGL(1,𝔽13) |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 2, 13]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^8);;
gap> SetConjugate(coll, 9, 2, F.9^8);;
gap> SetConjugate(coll, 9, 4, F.9^3);;
gap> SetConjugate(coll, 9, 5, F.9^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);