Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8101 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#8116 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#95 |
| commutator | C23 | C2×C42×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| prime radical | C24×C7 | C22×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C42×C7 ◁ G | |
562 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)(10,16,17,13)(11,15)(12,14)(18,22,23,20)(19,21)(24,25)(26,32)(27,30)(28,31)(29,34)(33,38)(35,37)(36,39);(8,10,11,18,12,17,19,23)(9,13,15,20,14,16,21,22)(24,26)(25,29,31,32)(27,35)(28,34)(30,38,39,37)(33,36);(24,27,28,36)(25,30,31,39)(26,33,34,35)(29,37,32,38)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1104);