Finite Group DatabasebetaFinite Group Databasebeta| order | 1260 |
| isoclinism | isoclinic class of D315 |
| automorphism | Aut(G) order 90720 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D630 |
| direct product | C2×D315 |
| unambiguous semidirect product | (C2×C5)⋊D63 |
| unambiguous semidirect product | (C2×C7)⋊D45 |
| unambiguous semidirect product | (C2×C9)⋊D35 |
| Fitting | C2×C5×C7×C9 ⋊ C2 |
| derived, Hall | C5×C7×C9 ⋊ C22 |
| Hall | C7×C9 ⋊ C2×D5 |
| Hall | C5×C9 ⋊ C2×D7 |
| Hall | C5×C7 ⋊ C2×D9 |
| Sylow | C9 ⋊ C2×D35 |
| Sylow | C7 ⋊ C2×D45 |
| Sylow | C5 ⋊ C2×D63 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D315 |
| commutator, cosocle, hypocenter | C5×C7×C9 | C22 |
| Frattini | C3 | C2×D105 |
| Fitting | C2×C5×C7×C9 | C2 |
| socle | C2×C3×C5×C7 | D3 |
| LCS | G ▷ C5×C7×C9 | |
| UCS | C1 ◁ C2 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 630 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2)(3,4,5,6,7,8,9,10,11)(12,13,14,15,16)(17,18,19,20,21,22,23);(4,11)(5,10)(6,9)(7,8)(13,16)(14,15)(18,23)(19,22)(20,21)318 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D11×D315, C2×C11×D315, C2×D3465, … |
| maximal normal subgroup of | C2×C11×D315, C2×C13×D315, C2×C17×D315, … |
| maximal quotient of | D11×D315, C2×C11×D315, C2×D3465, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1260, 39);