Finite Group DatabasebetaFinite Group Databasebeta| order | 5040 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, socle, derived, Hall | C32×C5×C7 ⋊ C22×C4 |
| Hall | C32×C7 ⋊ C22×AGL(1,𝔽5) |
| Hall | C32×C5 ⋊ C2×C4×D7 |
| Hall | C5×C7 ⋊ C4×D32 |
| Sylow | C32 ⋊ C2×D7×AGL(1,𝔽5) |
| Sylow | C7 ⋊ D32×AGL(1,𝔽5) |
| Sylow | C5 ⋊ #11759419 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32×C5×C7 | C22×C4 |
| prime radical | D5×(C3⋊D3)⋊D7 | C2 |
| cosocle | C32×C7×D5 | C23 |
| LCS | G ▷ C32×C5×C7 | |
| UCS | C1 | |
464 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^4,3^2,5,7|c^2=d,e^b=e^2,f^a=f^2,g^c=g^3,g^d=g^4,h^a=h^6,h^b=h^6,h^c=h^6117 conjugacy classes — view the full character table.
| maximal subgroup of | #72218138, D32×D7×AGL(1,𝔽5) |
| maximal normal subgroup of | #72218138, D32×D7×AGL(1,𝔽5) |
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, 7]);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 3, F.7^3);;
gap> SetConjugate(coll, 7, 4, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 8, 2, F.8^6);;
gap> SetConjugate(coll, 8, 3, F.8^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);