Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, Hall | C7×#2846 ⋊ C3 |
| Sylow | #2846 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ AΣL(1,𝔽64) |
1254 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 1254 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 19 points:
(2,4)(7,8,11)(9,10,12)(14,15,16)(17,19,18);(1,2)(3,4)(5,12,11,6)(7,10,8,9)(13,14,15,17,16,18,19)Polycyclic presentation:
2^2,3,2^5,7|b^a=bd,e^a=eg,e^c=ef,f^a=fh,f^c=e,g^c=gh,h^c=g,i^c=i^2116 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×AΣL(1,𝔽64) |
| maximal normal subgroup of | ΓL(1,𝔽8)×AΣL(1,𝔽64) |
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", "i");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 2, 2, 7]);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.5);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 3, F.7);;
gap> SetConjugate(coll, 9, 3, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);