Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#17319 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C7 | C24 |
| commutator | C22 | C22×C42×C7 |
| Frattini, cosocle | C42 | C24×C7 |
| socle | C22×C7 | C22×C42 |
| prime radical | C24×C7 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42×C7 ◁ G | |
834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7)(9,12)(13,15)(16,17,19,23,20,22,26,29)(18,21,25,28,24,27,30,31);(8,9,10,13,11,12,14,15)(16,18)(17,21)(19,25)(20,24)(22,27)(23,28)(26,30)(29,31);(18,24)(21,27)(25,30)(28,31);(17,22)(21,27)(23,29)(28,31)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 10307);