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×#8122 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#95 |
| commutator | C23 | C2×C42×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C24×C7 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C42×C7 ◁ G | |
520 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)(10,17,19,14)(11,21)(12,15)(13,16)(18,22,20,23)(24,25,28,32,29,33,27,31)(26,34,37,38,36,30,35,39);(8,10,12,20,13,19,11,18)(9,14,15,22,16,17,21,23)(24,26)(25,30,33,34)(27,35)(28,37)(29,36)(31,38,32,39);(8,11,13,12)(9,15,16,21)(10,18,19,20)(14,22,17,23)(24,27,29,28)(25,31,33,32)(26,35,36,37)(30,38,34,39)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1110);