Finite Group DatabasebetaFinite Group Databasebeta| order | 4000 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | 5−1+2 ⋊ #100 |
192 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 200 |
Polycyclic presentation:
2^2,5,2^3,5^2|a^2=e,e^2=f,g^5=h,b^a=bd,d^a=df,e^b=ef,g^a=g^4h^4,g^c=gh^3,h^a=h^4175 conjugacy classes — view the full character table.
| maximal subgroup of | #100×D25⋊C5, #72275822 |
| maximal normal subgroup of | #100×D25⋊C5, #72275822 |
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");;
gap> coll := SingleCollector(F, [2, 2, 5, 2, 2, 2, 5, 5]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^4*F.8^4);;
gap> SetConjugate(coll, 7, 3, F.7*F.8^3);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);