Finite Group DatabasebetaFinite Group Databasebeta| order | 3600 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C5 |
No aliases for this group.
| Hall | C32×C52 ⋊ C22⋊C4 |
| Sylow | C52 ⋊ #3722 |
| Sylow | C32 ⋊ C22⋊#1044 |
528 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 20 points:
(2,4)(5,6,9,7,8)(10,11,12,14,13)(16,17)(18,19);(1,2)(3,4)(6,7,8,9)(11,12,13,14)(15,16,17)(19,20)Polycyclic presentation:
2^4,3^2,5^2|a^2=c,b^a=bd,e^a=e^2,e^b=e^2,f^a=e^2f,f^b=f^2,g^a=g^2,g^c=g^4,h^a=g^3h^3,h^c=h^4102 conjugacy classes — view the full character table.
| maximal subgroup of | #72224629, #72224630, #72224631, … |
| maximal normal subgroup of | #72224629, #72224630, #72224631, … |
| maximal quotient of | C2×#72224607 |
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, 2, 2, 3, 3, 5, 5]);;
gap> SetPower(coll, 1, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.7^3*F.8^3);;
gap> SetConjugate(coll, 8, 3, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);