Finite Group DatabasebetaFinite Group Databasebeta| order | 3136 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C24×C72 ⋊ C22 |
| Sylow | C72 ⋊ C2×#120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D72 |
| commutator, socle, cosocle | C22×C72 | C24 |
| Fitting | C24×C72 | C22 |
| hypercenter | C24 | D72 |
| hypocenter | C72 | C2×#120 |
| LCS | G ▷ C22×C72 ▷ C72 | |
| UCS | C1 ◁ C22 ◁ C24 | |
2446 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 2446 conjugacy classes of subgroups — too many to draw. Only its 172 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,7^2|c^a=ce,d^a=def,g^a=g^6,g^b=g^6,h^a=g^2h,h^b=h^6256 conjugacy classes — view the full character table.
| maximal subgroup of | #72215039, #72215052, #72215058, … |
| maximal normal subgroup of | #72215039, #72215052, #72215058, … |
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, 2, 7, 7]);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 7, 2, F.7^6);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 2, F.8^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);