Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#9034 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #120 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | #386 |
| prime radical | C24×C7 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
482 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)(8,9,12,16,13,17,11,15)(10,18,19,14,21,22,20,23)(24,25)(26,34,36,38)(27,37,30,31)(28,32)(29,33)(35,39);(8,10)(9,14)(11,19)(12,20)(13,21)(15,22)(16,18)(17,23)(24,26,28,37,29,36,39,31)(25,30,32,38,33,27,35,34);(8,11,13,12)(9,15,17,16)(10,19,21,20)(14,22,23,18)(24,27,29,30)(25,31,33,37)(26,35,36,32)(28,34,39,38)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 2022);