Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#8468 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #379 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| prime radical | C7×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,16,18,13)(11,19,20,14)(12,15)(17,21)(22,23)(25,29,31,34)(30,38,39,37);(8,10)(9,13)(11,17)(12,18)(14,21)(15,16)(19,22)(20,23)(24,25)(26,30)(27,34)(28,31)(29,36)(32,37)(33,39)(35,38);(8,11,12,20)(9,14,15,19)(10,17,18,23)(13,21,16,22)(24,26,28,33)(25,30,31,39)(27,35,36,32)(29,37,34,38)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1456);