Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C7×#568 ⋊ C2×C3 |
| Hall | #11744235 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #3585 |
| Sylow | C7 ⋊ C3×#3585 |
2928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,7)(3,4,6)(9,15)(10,16)(13,20)(17,22);(9,15)(13,20)(14,21)(19,23);(8,10)(9,14)(11,17)(12,16)(13,19)(15,21)(18,22)(20,23);(2,3)(4,6)(5,7)(9,13)(11,18)(14,19)(15,20)(17,22)(21,23);(8,9,11,13,12,15,18,20)(10,14,17,19,16,21,22,23)Polycyclic presentation:
2^5,3,2^2,7|a^2=g,g^2=h,b^a=bgh,d^a=dh,e^a=eh,e^c=eh,g^b=gh,i^b=i^6,i^f=i^4157 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#3585 |
| maximal normal subgroup of | AGL(1,𝔽7)×#3585 |
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, 2, 3, 2, 2, 7]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.9^6);;
gap> SetConjugate(coll, 9, 6, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);