Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×#492 ⋊ C2×C3 |
| Hall | #11739219 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #3291 |
| Sylow | C7 ⋊ C3×#3291 |
920 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 920 conjugacy classes of subgroups — too many to draw. Only its 209 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 — 4 generators on 23 points:
(1,2,7)(3,4,6)(8,10,11,18,12,19,20,23)(9,14,16,22,15,21,13,17);(9,13,15,16)(14,17,21,22);(9,13)(10,18)(11,20)(15,16)(17,22)(19,23);(2,3)(4,6)(5,7)(8,9,12,15)(10,17,19,22)(11,13,20,16)(14,18,21,23)Polycyclic presentation:
2^4,3,2^3,7|a^2=h,c^2=f,d^2=g,g^2=h,b^a=bg,c^a=cg,c^b=cf,d^a=dgh,d^b=dg,f^a=fh,g^a=gh,g^b=gh,i^a=i^6,i^e=i^4109 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, 2, 3, 2, 2, 2, 7]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*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, 5, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);