Finite Group DatabasebetaFinite Group Databasebeta| order | 5040 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, socle, derived, Hall | C32×C5×C7 ⋊ C2×C8 |
| Hall | C32×C7 ⋊ D5⋊C8 |
| Hall | C32×C5 ⋊ C8×D7 |
| Hall | C5×C7 ⋊ C2×AGL(1,𝔽9) |
| Sylow | C32 ⋊ #10587920 |
| Sylow | C7 ⋊ D5⋊AGL(1,𝔽9) |
| Sylow | C5 ⋊ D7×AGL(1,𝔽9) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32×C5×C7 | C2×C8 |
| prime radical | C3⋊D3×D35 | C4 |
| cosocle | C7×#4379 | C22 |
| LCS | G ▷ C32×C5×C7 | |
| UCS | C1 | |
108 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 108 conjugacy classes of subgroups — too many to draw. Only its 25 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|a^2=c,c^2=d,e^a=e^2f^2,e^c=e^2f,e^d=e^2,f^a=e^2,f^c=ef,f^d=f^2,g^a=g^3,g^b=g^4,g^c=g^4,h^b=h^669 conjugacy classes — view the full character table.
| maximal subgroup of | #72217456, D7×D5⋊AGL(1,𝔽9) |
| maximal normal subgroup of | #72217456, D7×D5⋊AGL(1,𝔽9) |
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, 1, F.3);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 5, 3, F.5^2*F.6);;
gap> SetConjugate(coll, 5, 4, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.5^2);;
gap> SetConjugate(coll, 6, 3, F.5*F.6);;
gap> SetConjugate(coll, 6, 4, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^3);;
gap> SetConjugate(coll, 7, 2, F.7^4);;
gap> SetConjugate(coll, 7, 3, F.7^4);;
gap> SetConjugate(coll, 8, 2, F.8^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);