Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C7 · Schur multiplier (H2) = C24×C4 |
| direct product | C7×#20437 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C7×C8 | C23 |
| commutator | C22 | C22×C42×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C23×C7 | C23×C4 |
| prime radical | C24×C7 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7×C8 ◁ G | |
1166 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)(8,9,10,11)(16,17,21,24,20,23,29,19)(18,25,28,26,27,22,30,31);(9,11)(12,13,14,15)(16,18)(17,22)(19,26)(20,27)(21,28)(23,25)(24,31)(29,30);(16,19,29,23,20,24,21,17)(18,26,30,25,27,31,28,22);(18,27)(22,25)(26,31)(28,30)784 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13425);