Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×C4×C52 ⋊ C2×C4 |
| Hall | C3×C52 ⋊ C4×D4 |
| Sylow | C52 ⋊ C4×D12 |
| Sylow | C3 ⋊ #11717363 |
436 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 436 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 120 |
Polycyclic presentation:
2^5,3,5^2|b^2=d,c^2=e,c^a=ce,f^a=f^2,g^a=g^4,g^b=g^3,g^d=g^4,h^a=g^3h,h^b=h^3,h^d=h^478 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#11717363, #72230582, #72230588, … |
| maximal normal subgroup of | D3×#11717363, #72230582, #72230588, … |
| maximal quotient of | #72260023 |
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, 3, 5, 5]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> SetConjugate(coll, 7, 2, F.7^3);;
gap> SetConjugate(coll, 7, 4, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.7^3*F.8);;
gap> SetConjugate(coll, 8, 2, F.8^3);;
gap> SetConjugate(coll, 8, 4, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);