Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#9549 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #121 |
| commutator | C2×C4 | C22×C7×C8 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | #388 |
| prime radical | C23×C7 | #117 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,11,14,12,10,13,15)(17,21)(18,24)(19,26)(22,29)(23,28)(27,30);(8,10,11,15,12,9,13,14)(16,17,19,21,20,23,26,28)(18,22,25,27,24,30,31,29);(16,18,19,25,20,24,26,31)(17,22,21,27,23,30,28,29)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 2537);