Finite Group DatabasebetaFinite Group Databasebeta| order | 3200 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C52×#117 ⋊ C22 |
| Sylow | C52 ⋊ #3021 |
1806 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 1806 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 26 points:
(3,10)(6,15)(11,12)(14,16)(17,18,20,21,19)(22,23,25,26,24);(2,7)(4,13)(6,14)(8,9)(11,12)(15,16)(18,19)(20,21)(23,24)(25,26);(1,3)(2,6)(4,12)(5,10)(7,16)(8,15)(9,14)(11,13)(18,19)(20,21)(23,24)(25,26);(1,2,5,8)(3,6,10,15)(4,7,13,9)(11,14,12,16)(23,24)(25,26)Polycyclic presentation:
2^7,5^2|a^2=g,f^2=g,c^a=cf,c^b=ce,d^b=dg,e^a=eg,f^a=fg,f^b=fg,f^c=fg,h^a=h^4,h^b=h^4,h^c=h^4,i^a=h^3i,i^b=i^4,i^c=i^4197 conjugacy classes — view the full character table.
| maximal subgroup of | #72226522 |
| maximal normal subgroup of | #72226522 |
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, 2, 2, 5, 5]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> SetConjugate(coll, 8, 2, F.8^4);;
gap> SetConjugate(coll, 8, 3, F.8^4);;
gap> SetConjugate(coll, 9, 1, F.8^3*F.9);;
gap> SetConjugate(coll, 9, 2, F.9^4);;
gap> SetConjugate(coll, 9, 3, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);