Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#22690 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | D42 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ G | |
846 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)(8,9,10,11)(12,13)(14,26)(15,30)(16,22)(17,24)(18,23)(19,25)(20,28)(21,31)(27,41)(29,39)(32,40)(33,43)(34,37)(35,38)(36,42);(12,14,19,29)(13,20,25,38)(15,16,32,34)(17,35,18,28)(21,37,41,22)(23,39,24,26)(27,36,31,33)(30,43,40,42);(12,15,18,31,19,32,17,27)(13,21,24,30,25,41,23,40)(14,16,28,33,29,34,35,36)(20,37,26,43,38,22,39,42);(12,16)(13,22)(14,27)(15,28)(17,36)(18,33)(19,34)(20,30)(21,39)(23,43)(24,42)(25,37)(26,41)(29,31)(32,35)(38,40)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 15678);