Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#13154 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #442 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C23×C7 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C7×#97 ◁ G | |
314 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)(8,9)(10,11)(12,13,17,22,18,24,36,23)(14,25,27,20,29,38,28,39)(15,30,31,42,33,19,32,37)(16,34,35,21,26,40,43,41);(12,14,26,15,17,28,43,32,18,29,16,33,36,27,35,31)(13,19,21,38,23,37,34,39,24,30,41,25,22,42,40,20);(9,11)(12,15,16,28,17,32,35,29,18,33,26,27,36,31,43,14)(13,20,21,30,23,38,34,42,24,39,41,19,22,25,40,37)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6142);