Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#8118 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #95 |
| commutator | C23 | C2×C42×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | C2×#95 |
| prime radical | C24×C7 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C42×C7 ◁ G | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,18,20,14)(11,15)(12,17)(13,16)(19,23,21,22)(24,25,27,30,28,31,35,38)(26,32,34,39,33,29,36,37);(8,10,12,21,13,20,11,19)(9,14,17,23,16,18,15,22)(24,26)(25,29,31,32)(27,34)(28,33)(30,37,38,39)(35,36);(8,11,13,12)(9,15,16,17)(10,19,20,21)(14,22,18,23)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1106);