Finite Group DatabasebetaFinite Group Databasebeta| order | 7056 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Hall | #11716072 ⋊ C32 |
| Hall | ΓL(1,𝔽8)2 ⋊ SD16 |
| Sylow | C72 ⋊ C32×SD16 |
280 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
Polycyclic presentation:
2^2,3^2,2^2,7^2|a^2=e,e^2=f,b^a=bef,e^b=ef,g^a=g^6,g^c=h,g^d=g^2,h^a=h^6,h^c=g^5h^3,h^d=h^2103 conjugacy classes — view the full character table.
| maximal subgroup of | SD16×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72245389, #72245395 |
| maximal normal subgroup of | SD16×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72245389, #72245395 |
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, 3, 3, 2, 2, 7, 7]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*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.8);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 8, 3, F.7^5*F.8^3);;
gap> SetConjugate(coll, 8, 4, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);