Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24×C3 |
No aliases for this group.
| Fitting | C23×C32×C7 ⋊ C22 |
| Hall | C32×C7 ⋊ #120 |
| Sylow | C32 ⋊ #6500 |
| Sylow | C7 ⋊ #64146 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D7×C3⋊D3 |
| commutator, socle, cosocle | C22×C32×C7 | C23 |
| Fitting | C23×C32×C7 | C22 |
| hypercenter | C23 | D7×C3⋊D3 |
| hypocenter | C32×C7 | #120 |
| LCS | G ▷ C22×C32×C7 ▷ C32×C7 | |
| UCS | C1 ◁ C22 ◁ C23 | |
780 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 780 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 504 |
Polycyclic presentation:
2^5,3^2,7|b^a=bd,c^a=ce,f^c=f^2,g^c=g^2,h^b=h^6180 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×#6500, D7×#64146, #72246776, … |
| maximal normal subgroup of | C3⋊D3×#6500, D7×#64146, #72246776, … |
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, 3, 7]);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 8, 2, F.8^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);