Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#13173 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #466 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1942 |
| prime radical | C23×C7 | C2×Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
262 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)(8,9,12,20,15,21,14,18)(10,22,26,39,27,44,24,16)(11,28,30,60,33,61,32,59)(13,35,34,46,36,45,29,19)(17,40,42,63,43,62,41,31)(23,49,53,68,54,69,51,58)(25,55,56,67,57,38,50,48)(37,52,47,65,64,71,66,70);(8,10,12,26,15,27,14,24)(9,16,20,22,21,39,18,44)(11,23,30,53,33,54,32,51)(13,25,34,56,36,57,29,50)(17,37,42,47,43,64,41,66)(19,38,35,48,46,55,45,67)(28,58,60,49,61,68,59,69)(31,52,40,65,63,71,62,70);(8,11,29,62,14,32,36,63,15,33,34,40,12,30,13,31)(9,17,35,60,20,42,46,61,21,43,45,59,18,41,19,28)(10,23,50,70,24,51,57,71,27,54,56,65,26,53,25,52)(16,37,48,49,22,47,55,68,39,64,67,69,44,66,38,58)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6161);