Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| isoclinism | isoclinic class of #4245 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#66675 |
| Hall | C3×C5×C7 ⋊ #41 |
| Hall | C5×C7 ⋊ #235 |
| Hall | C3×C7 ⋊ C5×#41 |
| Hall | C3×C5 ⋊ #1157 |
| Sylow | C7 ⋊ C5×#235 |
| Sylow | C3 ⋊ C5×#1157 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C5 | D3×D7 |
| commutator, cosocle | C2×C3×C7 | C23×C5 |
| Frattini | C2 | C2×C5×D3×D7 |
| Fitting | C3×C4×C5×C7 | C22 |
| socle | C2×C3×C5×C7 | C23 |
| hypocenter | C3×C7 | C5×#41 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C4×C5 | |
160 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 840 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5)(6,7,9,11,12,10,8)(13,14,15)(16,18,19,22)(17,21,20,23);(14,15)(17,20)(21,23);(7,8)(9,10)(11,12)(16,17)(18,21)(19,20)(22,23)270 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 685);