Finite Group DatabasebetaFinite Group Databasebeta| order | 16224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Hall | #72259202 ⋊ C3 |
| Hall | #92788 ⋊ C2×M16 |
| Sylow | C132 ⋊ C2×C3×M16 |
596 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 596 conjugacy classes of subgroups — too many to draw. Only its 78 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:
2^3,3,2^2,13^2|a^2=e,c^2=f,e^2=f,b^a=bf,g^a=g^5,g^d=g^3,g^e=g^12,h^a=g^6h^8,h^d=h^3,h^e=h^12172 conjugacy classes — view the full character table.
| maximal subgroup of | #72259347, #72259572, #72259595 |
| maximal normal subgroup of | #72259347, #72259572, #72259595 |
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, 3, 2, 2, 13, 13]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^5);;
gap> SetConjugate(coll, 7, 4, F.7^3);;
gap> SetConjugate(coll, 7, 5, F.7^12);;
gap> SetConjugate(coll, 8, 1, F.7^6*F.8^8);;
gap> SetConjugate(coll, 8, 4, F.8^3);;
gap> SetConjugate(coll, 8, 5, F.8^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);