Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1366 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#11956 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #333 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C2×C7 | C2×#333 |
| prime radical | C23×C7 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,16,25,17,35,33,13,18,27,34,11,15,24,31)(10,19,22,32,20,30,26,38,21,36,37,39,23,14,28,29);(8,10,13,21)(9,14)(11,20,25,23)(12,26,27,28)(15,19)(16,32)(17,36)(18,30)(22,24,37,35)(29,33)(31,38)(34,39);(8,11,13,25)(9,15,18,17)(10,20,21,23)(12,24,27,35)(14,19,30,36)(16,31,34,33)(22,26,37,28)(29,32,38,39)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4944);