Finite Group DatabasebetaFinite Group Databasebeta| order | 2430 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C32 |
No aliases for this group.
| Fitting, Hall | C5×3−1+4 ⋊ C2 |
| derived | C3×C5×C9 ⋊ C2×C32 |
| Sylow | 3−1+4 ⋊ D5 |
| Sylow | C5 ⋊ #4207⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C5×C9 | C2×C32 |
| Frattini | C3 | C32×C3⋊D15 |
| Fitting | C5×3−1+4 | C2 |
| socle | C3×C5 | C32×C3⋊D3 |
| LCS | G ▷ C3×C5×C9 | |
| UCS | C1 | |
362 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 135 |
Polycyclic presentation:
2,3^4,5,3|e^3=g,c^a=cg^2,d^a=d^2,d^b=dg^2,e^a=e^2g^2,e^b=eg,e^c=eg,f^a=f^4,g^a=g^2221 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#4207⋊C3, #72247325, #72247520, … |
| maximal normal subgroup of | D5×#4207⋊C3, #72247520, #72247526 |
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");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 3, 5, 3]);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.7^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);