Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#10033 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | #392 |
| prime radical | C7×#3100 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,12,10)(11,14,13,15)(16,17)(18,24)(19,27)(20,23)(21,26)(22,29)(25,31)(28,30);(9,14)(10,15)(11,13)(21,30)(24,31);(8,10)(9,12)(11,15)(13,14)(16,18)(17,21)(19,28)(20,25)(22,31)(23,30)(24,27)(26,29)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 3021);