Finite Group DatabasebetaFinite Group Databasebeta| order | 5408 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C132 ⋊ C2×M16 |
298 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 54 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:
2^5,13^2|a^2=d,d^2=e,b^a=be,f^a=f^12,f^c=f^12,g^a=f^2g,g^c=g^12428 conjugacy classes — view the full character table.
| maximal subgroup of | #72259204, #72259277, #72259282, … |
| maximal normal subgroup of | #72259204, #72259277, #72259282, … |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 13, 13]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 6, 3, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.6^2*F.7);;
gap> SetConjugate(coll, 7, 3, F.7^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);