Finite Group DatabasebetaFinite Group Databasebeta| order | 1260 |
| isoclinism | isoclinic class of C22⋊ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C9 · Schur multiplier (H2) = C2 |
| derived, Hall | C22×C7 ⋊ C5×C9 |
| Hall | C22×C5×C7 ⋊ C9 |
| Hall | C5×C7 ⋊ C22⋊C9 |
| Hall | C22×C5 ⋊ C7⋊C9 |
| Sylow | C7 ⋊ C5×C22⋊C9 |
| Sylow | C22 ⋊ C5×C7⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C5 | C22⋊ΓL(1,𝔽8) |
| commutator, hypocenter | C22×C7 | C5×C9 |
| Frattini | C3 | C5×C22⋊ΓL(1,𝔽8) |
| Fitting, socle, prime radical | C22×C3×C5×C7 | C3 |
| cosocle | C22×C3×C7 | C3×C5 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 ◁ C3×C5 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 630 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4,5)(7,8,9)(11,12,13)(14,16,15)(17,18,20,19,21,23,22,24,25);(6,9)(7,8)(10,11,12,14,13,15,16)180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1260, 32);