Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#7926 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C22×C7 | C2×#95 |
| prime radical | C24×C7 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C42×C7 ◁ G | |
594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,16,18,13)(11,14)(12,15)(17,22,23,20)(19,21)(24,25,27,31,29,33,28,32)(26,30,35,38,34,37,36,39);(8,10)(9,13)(11,17)(12,18)(14,20)(15,16)(19,23)(21,22)(24,26,28,36,29,34,27,35)(25,30,32,39,33,37,31,38);(8,11,12,19)(9,14,15,21)(10,17,18,23)(13,20,16,22)(26,34)(30,37)(35,36)(38,39)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 914);