Finite Group DatabasebetaFinite Group Databasebeta| order | 3136 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C72 ⋊ C2×#100 |
726 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 726 conjugacy classes of subgroups — too many to draw. Only its 74 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|a^2=d,d^2=f,e^2=f,b^a=bdef,d^b=df,e^a=ef,e^b=ef,g^a=g^6,g^c=g^6,h^a=g^2h,h^c=h^6178 conjugacy classes — view the full character table.
| maximal subgroup of | #72214932, #72214953, #72245889, … |
| maximal normal subgroup of | #72214932, #72214953, #72245889, … |
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> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 7, 3, F.7^6);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 3, F.8^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);