Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#13126 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | #124 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1924 |
| prime radical | C2×C7×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C22×C7×C8 ◁ G | |
300 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)(8,9,13,19,14,20,37,52,15,21,38,53,12,18,23,40)(10,22,27,54,25,44,55,50,28,42,35,66,26,16,39,43)(11,29,33,62,32,49,56,51,34,47,36,67,31,17,45,48)(24,57,58,69,30,63,64,71,59,41,61,68,60,46,65,70);(8,10,23,55,12,26,38,27,15,28,37,39,14,25,13,35)(9,16,40,54,18,42,53,43,21,44,52,66,20,22,19,50)(11,30,45,61,31,24,36,64,34,60,56,58,32,59,33,65)(17,46,67,69,47,41,51,70,49,63,62,68,29,57,48,71);(8,11,12,31,15,34,14,32)(9,17,18,47,21,49,20,29)(10,24,25,30,28,59,26,60)(13,36,23,56,38,33,37,45)(16,41,22,46,44,57,42,63)(19,51,40,62,53,48,52,67)(27,61,55,65,35,58,39,64)(43,69,54,71,50,68,66,70)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6114);