Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#21430 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C7 | C23 |
| commutator | C22 | C23×C7×C8 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C23×C7 | C23×C4 |
| prime radical | C24×C7 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C7 ◁ G | |
858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,17)(18,21)(19,20)(22,23)(24,25,29,26,28,31,27,30);(9,12)(13,15)(24,26,29,31,28,30,27,25);(8,10,11,14)(9,13,12,15)(17,20)(21,23)(24,27,28,29)(25,30,31,26);(16,18,19,22)(17,21,20,23)784 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 14418);