Finite Group DatabasebetaFinite Group Databasebeta| order | 3200 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C5 |
No aliases for this group.
| Fitting | C52×H(5,𝔽2) ⋊ C4 |
| Sylow | C52 ⋊ #1792 |
2138 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 2138 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,3)(2,5)(4,6)(7,8)(9,10,13,11,12)(14,15,16,18,17);(4,7)(6,8);(1,2,4,6,5,3,7,8)(10,11,12,13)(15,16,17,18)Polycyclic presentation:
2^7,5^2|a^2=d,d^2=g,e^2=g,b^a=bf,c^a=ce,d^b=dg,d^c=dg,e^b=eg,e^c=eg,f^a=fg,f^c=fg,h^a=h^2,h^d=h^4,i^a=h^3i^3,i^d=i^4134 conjugacy classes — view the full character table.
| maximal subgroup of | #72226461 |
| maximal normal subgroup of | #72226461 |
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, 4, F.7);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, 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);