Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#21296 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C22×C7 | C2×C4×D4 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 47 points:
(1,2,3,4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,17,21,25,22,26,20,35)(18,23,29,37,30,38,28,44)(19,31,34,39,32,45,33,24)(27,40,43,46,41,47,42,36);(9,12)(13,15)(16,18,21,29,22,30,20,28)(17,23,25,37,26,38,35,44)(19,27,34,43,32,41,33,42)(24,36,31,40,39,46,45,47);(16,19,20,33,22,32,21,34)(17,24,25,31,26,39,35,45)(18,27,28,42,30,41,29,43)(23,36,37,40,38,46,44,47);(8,10,11,14)(9,13,12,15)(19,32)(24,39)(27,41)(31,45)(33,34)(36,46)(40,47)(42,43)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 14284);