Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C4 |
| direct product | C7×#9585 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #121 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | #388 |
| prime radical | C23×C7 | #126 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(9,13)(10,16)(11,18)(14,21)(15,20)(19,22)(24,25,28,32,29,33,27,30)(26,34,36,31,37,38,35,39);(8,9,11,13,12,15,18,20)(10,14,17,19,16,22,23,21)(25,30,33,32)(31,34,39,38);(8,10,11,17,12,16,18,23)(9,14,13,19,15,22,20,21)(24,26,27,35,29,37,28,36)(25,31,32,38,33,39,30,34)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 2573);