Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C2×C7×#95 ⋊ C2×C3 |
| Hall | #11727180 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #1438 |
| Sylow | C7 ⋊ C3×#1438 |
2336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 23 points:
(6,8)(10,12)(14,16)(18,22,21,19,20,23);(1,3,4,2)(10,12)(13,14)(15,16)(17,18,21,22,20,23,19);(1,2,4,3)(5,6,7,8)(9,10)(11,12)(14,16)Polycyclic presentation:
2^3,3,2^4,7|a^2=g,b^2=f,b^a=be,c^a=cgh,c^b=cefh,i^c=i^6,i^d=i^4172 conjugacy classes — view the full character table.
| maximal subgroup of | #72276889, AGL(1,𝔽7)×#1438 |
| maximal normal subgroup of | #72276889, AGL(1,𝔽7)×#1438 |
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.7);;
gap> SetPower(coll, 2, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.8);;
gap> SetConjugate(coll, 9, 3, F.9^6);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);