Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#7476 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #367 |
| commutator | C16 | C42×C7 |
| Frattini, cosocle | C22×C16 | C22×C7 |
| socle | C2×C7 | #1380 |
| prime radical | C23×C7 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ C22×C7×C16 ◁ G | |
162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 2 generators on 71 points:
(1,2,3,4,5,6,7)(8,9,12,19,16,22,15,41)(10,23,25,55,28,57,34,56)(11,20,30,50,33,53,32,67)(13,36,35,52,31,18,38,49)(14,39,29,51,37,48,40,21)(17,42,43,24,47,59,46,61)(26,64,62,45,60,54,68,70)(27,66,63,44,65,69,58,71);(8,10,13,26,37,65,30,42,12,34,35,68,40,63,33,61,16,28,31,60,14,27,32,59,15,25,38,62,29,58,11,24)(9,17,20,44,51,70,49,56,41,43,67,69,39,64,18,23,22,47,53,71,21,45,52,55,19,46,50,66,48,54,36,57)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 464);