Finite Group DatabasebetaFinite Group Databasebeta| order | 3200 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4×C5 |
No aliases for this group.
| Fitting | C52×#124 ⋊ C4 |
| Sylow | C52 ⋊ #1910 |
1022 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 1022 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 26 points:
(2,8)(4,5)(7,14)(9,10)(11,13)(15,16)(17,18,21,19,20)(22,23,24,26,25);(1,3,6,12)(2,7,9,16)(4,11,5,13)(8,15,10,14);(1,2,4,8,6,9,5,10)(3,7,11,15,12,16,13,14)(18,19,20,21)(23,24,25,26)Polycyclic presentation:
2^7,5^2|a^2=d,b^2=g,d^2=g,f^2=g,c^a=cf,c^b=ce,d^c=dg,e^a=eg,f^b=fg,f^c=fg,h^a=h^2,h^d=h^4,i^a=h^3i^3,i^d=i^4113 conjugacy classes — view the full character table.
| maximal subgroup of | #72226451 |
| maximal normal subgroup of | #72226451 |
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.4);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 4, 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, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*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.8^2);;
gap> SetConjugate(coll, 8, 4, F.8^4);;
gap> SetConjugate(coll, 9, 1, F.8^3*F.9^3);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);