Finite Group DatabasebetaFinite Group Databasebeta| order | 3200 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22×C4×C52 ⋊ C2×C4 |
| Sylow | C52 ⋊ #1890 |
990 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 990 conjugacy classes of subgroups — too many to draw. Only its 87 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:
(1,2,5,3,4)(6,7,10,9,8)(11,14,15,21)(12,17,18,25)(13,19,20,26)(16,22,23,24);(2,3,4,5)(7,9,8,10)(11,13)(12,16)(14,19)(15,20)(17,24)(18,23)(21,26)(22,25);(7,8)(9,10)(11,12,15,18)(13,16,20,23)(14,17,21,25)(19,24,26,22)Polycyclic presentation:
2^7,5^2|a^2=g,b^2=d,c^2=g,c^a=cf,c^b=ce,e^a=eg,f^b=fg,h^a=h^4,h^b=h^3,h^d=h^4,i^a=h^2i,i^b=i^3,i^d=i^492 conjugacy classes — view the full character table.
| maximal subgroup of | #72226529 |
| maximal normal subgroup of | #72226529 |
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, 2, F.4);;
gap> SetPower(coll, 3, F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> SetConjugate(coll, 8, 2, F.8^3);;
gap> SetConjugate(coll, 8, 4, F.8^4);;
gap> SetConjugate(coll, 9, 1, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 2, F.9^3);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);