Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of #204 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C7 · Schur multiplier (H2) = C1 |
| direct product | C5×C7×Dic12 |
| Hall | C3×C5×C7 ⋊ Q16 |
| Hall | C3×C7 ⋊ C5×Q16 |
| Hall | C3×C5 ⋊ C7×Q16 |
| Sylow | C3 ⋊ C5×C7×Q16 |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 1680 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3,4,5)(6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20)(21,28,27,26,25,24,23,22)(29,30,31);(13,21,17,25)(14,22,18,26)(15,23,19,27)(16,24,20,28)(30,31)525 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 329);