Finite Group DatabasebetaFinite Group Databasebeta| order | 32448 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | D4×C132 ⋊ C2×C3×C4 |
| Hall | #72259273 ⋊ C3 |
| Hall | #92788 ⋊ C4×D8 |
| Sylow | C132 ⋊ C3×C4×D8 |
1100 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 1100 conjugacy classes of subgroups — too many to draw. Only its 92 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:
2^3,3,2^3,13^2|a^2=f,c^2=e,f^2=g,b^a=bf,f^b=fg,h^a=h^12,h^c=h^8,h^d=h^3,h^e=h^12,i^a=h^2i,i^c=i^8,i^d=i^3,i^e=i^12128 conjugacy classes — view the full character table.
| maximal subgroup of | #72259662 |
| maximal normal subgroup of | #72259662 |
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, 13, 13]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^12);;
gap> SetConjugate(coll, 8, 3, F.8^8);;
gap> SetConjugate(coll, 8, 4, F.8^3);;
gap> SetConjugate(coll, 8, 5, F.8^12);;
gap> SetConjugate(coll, 9, 1, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 3, 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);