Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#13122 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C2×D4 |
| commutator | C2×C4 | C22×C7×C8 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C22×C7 | C8×D4 |
| prime radical | C2×C7×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ C22×C7×C16 ◁ G | |
326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(9,10)(12,13,18,23,19,22,37,40,16,21,25,26,17,34,36,42)(14,24,29,41,30,35,39,43,27,15,31,33,28,32,20,38);(8,9,11,10)(12,14,25,31,19,30,36,20,16,27,18,29,17,28,37,39)(13,15,26,41,22,32,42,43,21,24,23,33,34,35,40,38);(8,10,11,9)(12,15,16,24)(13,20,21,39)(14,26,27,23)(17,35,19,32)(18,33,25,41)(22,29,34,31)(28,40,30,42)(36,43,37,38)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6110);