Finite Group DatabasebetaFinite Group Databasebeta| order | 2880 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C32×C4×C5 ⋊ C2×C4 |
| Hall | C32×C5 ⋊ #518 |
| Sylow | C32 ⋊ #66258 |
| Sylow | C5 ⋊ #10594996 |
1588 conjugacy classes of subgroups; only the 235 normal subgroups are shown.
This group has 1588 conjugacy classes of subgroups — too many to draw. Only its 235 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=e,b^a=bf,c^a=cf,d^b=df,g^a=g^2,g^b=g^2,g^d=g^2,h^a=g^2h,h^b=h^2,h^d=h^2,i^a=i^3,i^e=i^4132 conjugacy classes — view the full character table.
| maximal subgroup of | #72260215, AGL(1,𝔽5)×#64308, #72260255 |
| maximal normal subgroup of | #72260215, AGL(1,𝔽5)×#64308, #72260255 |
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.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 8, 4, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^3);;
gap> SetConjugate(coll, 9, 5, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);