Finite Group DatabasebetaFinite Group Databasebeta| order | 9360 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Fitting, socle, derived | C3×C5×C13 ⋊ C3×C42 |
| Hall | C3×C5×C13⋊C3 ⋊ C42 |
| Hall | C3×C13⋊C3 ⋊ C4×AGL(1,𝔽5) |
| Hall | C5×C13 ⋊ C3×C4×Dic3 |
| Sylow | C13 ⋊ C3×Dic3×AGL(1,𝔽5) |
| Sylow | C5 ⋊ Dic3×AGL(1,𝔽13) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C5×C13 | C3×C42 |
| prime radical | C3×D5×D13⋊C3 | C22 |
| cosocle | C3×D5×D13 | C22×C3 |
| LCS | G ▷ C3×C5×C13 | |
| UCS | C1 | |
272 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 195 |
Polycyclic presentation:
2^2,3,2^2,3,5,13|a^2=e,b^2=d,f^a=f^2,g^b=g^3,g^d=g^4,h^a=h^8,h^b=h^8,h^c=h^3,h^d=h^12,h^e=h^1299 conjugacy classes — view the full character table.
| maximal subgroup of | D3×AGL(1,𝔽5)×AGL(1,𝔽13) |
| maximal normal subgroup of | D3×AGL(1,𝔽5)×AGL(1,𝔽13) |
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, 3, 2, 2, 3, 5, 13]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.7^3);;
gap> SetConjugate(coll, 7, 4, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^8);;
gap> SetConjugate(coll, 8, 2, F.8^8);;
gap> SetConjugate(coll, 8, 3, F.8^3);;
gap> SetConjugate(coll, 8, 4, F.8^12);;
gap> SetConjugate(coll, 8, 5, F.8^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);