Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#23203 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | D42 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C22×C7×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ G | |
892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4,5,6,7)(8,9,11,10)(12,13,18,23,19,24,17,22)(14,25,29,20,30,37,28,38)(15,31,32,40,33,39,16,21)(26,42,34,41,27,35,43,36);(8,10,11,9)(12,14,19,30)(13,20,24,38)(15,26,33,27)(16,34,32,43)(17,29,18,28)(21,35,40,42)(22,37,23,25)(31,41,39,36);(12,15,17,16,19,33,18,32)(13,21,23,31,24,40,22,39)(14,26,29,34,30,27,28,43)(20,35,25,41,38,42,37,36);(12,16,17,33,19,32,18,15)(13,21,22,39,24,40,23,31)(14,27,28,34,30,26,29,43)(20,36,25,35,38,41,37,42)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16191);