Finite Group DatabasebetaFinite Group Databasebeta| order | 2880 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C32×C5×D4 ⋊ C2×C4 |
| Hall | C32×C5 ⋊ C2×#102 |
| Sylow | C32 ⋊ C2×#4007 |
| Sylow | C5 ⋊ #10591416 |
1044 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 1044 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 120 |
Polycyclic presentation:
2^6,3^2,5|a^2=d,d^2=f,e^2=f,b^a=be,d^b=df,e^b=ef,g^a=g^2,g^c=g^2,h^a=g^2h,h^c=h^2,i^a=i^3,i^d=i^496 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×#63930, AGL(1,𝔽5)×#63926, #72230434, … |
| maximal normal subgroup of | AGL(1,𝔽5)×#63930, AGL(1,𝔽5)×#63926, #72230434, … |
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, 3, 3, 5]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^3);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);