Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C7×#493 ⋊ C2×C3 |
| Hall | #11729049 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #2213 |
| Sylow | C7 ⋊ C3×#2213 |
340 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,7)(3,4,6)(9,14,16,17)(13,15,21,22);(9,13)(10,18)(11,19)(14,22)(15,17)(16,21)(20,23);(2,3)(4,6)(5,7)(8,9,12,16)(10,13,20,21)(11,14,19,17)(15,23,22,18)Polycyclic presentation:
2^3,3,2^4,7|a^2=h,c^2=f,e^2=gh,g^2=h,b^a=be,c^a=cg,c^b=cf,e^a=eg,e^b=eg,f^a=fh,g^a=gh,g^b=gh,i^a=i^6,i^d=i^491 conjugacy classes — view the full character table.
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, 2, 3, 2, 2, 2, 2, 7]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 5, F.7*F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^6);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);