Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#61334 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| prime radical | C7×#3506 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C7×#533 ◁ G | |
1794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3,4,5,6,7)(9,12)(13,14)(17,22)(18,25)(19,21)(23,30)(26,28)(29,31);(8,9,11,12)(10,14,15,13)(16,17,20,22)(18,23,26,29)(19,27,21,24)(25,31,28,30);(16,18,20,26)(17,23,22,29)(19,25,21,28)(24,30,27,31);(8,10,11,15)(9,13,12,14)(16,19,20,21)(17,24,22,27)(18,25,26,28)(23,30,29,31);(17,22)(23,29)(24,27)(30,31)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54322);