Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#19596 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #127 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| socle | C22×C7 | C2×#127 |
| prime radical | C7×#3147 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
814 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)(9,12)(10,14)(13,15)(16,17)(18,25)(19,23)(20,29)(21,24)(22,27)(26,30)(28,31);(16,18,20,27,21,28,19,26)(17,22,23,25,24,30,29,31);(8,9,10,12,11,13,14,15);(16,19,21,20)(17,23,24,29)(18,26,28,27)(22,25,30,31)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12584);