Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#13157 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1936 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C23×C7 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 71 points:
(1,2,3,4,5,6,7)(8,9,14,21,16,22,15,44)(10,23,27,46,28,55,42,56)(11,29,33,51,34,57,43,65)(12,35,13,39,38,20,37,19)(17,26,47,25,49,40,48,36)(18,31,52,32,54,66,53,41)(24,59,62,70,63,45,30,50)(58,67,68,61,71,60,69,64);(8,10,12,36,15,27,37,26,16,28,38,25,14,42,13,40)(9,17,39,23,21,48,20,56,22,49,19,55,44,47,35,46)(11,30,66,61,33,63,41,67,34,62,31,64,43,24,32,60)(18,50,51,69,53,59,29,58,54,70,65,68,52,45,57,71);(8,11,12,31,15,33,37,32,16,34,38,66,14,43,13,41)(9,18,19,51,21,53,35,29,22,54,39,65,44,52,20,57)(10,24,25,60,27,30,40,61,28,63,36,67,42,62,26,64)(17,45,23,58,48,50,56,68,49,59,55,71,47,70,46,69)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6145);