Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#13195 |
| 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×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ C22×C7×C16 ◁ G | |
438 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)(9,11)(12,13,18,24,19,23,33,39,16,22,26,21,17,34,15,32)(14,25,30,36,31,40,42,43,28,20,35,38,29,37,27,41);(8,9,10,11)(12,14,26,35,19,31,15,27,16,28,18,30,17,29,33,42)(13,20,21,36,23,37,32,43,22,25,24,38,34,40,39,41);(12,15,17,26,16,33,19,18)(13,21,23,32,22,24,34,39)(14,27,29,35,28,42,31,30)(20,36,37,43,25,38,40,41)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6183);