Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#8056 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C22×C7 | C2×#95 |
| prime radical | C24×C7 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C42×C7 ◁ G | |
524 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)(8,9,11,16,13,17,12,15)(10,18,20,14,19,23,21,22)(24,25)(26,32)(27,30)(28,31)(29,34)(33,38)(35,37)(36,39);(8,10,12,21,13,19,11,20)(9,14,16,23,17,22,15,18)(24,26)(25,29,31,32)(27,33)(28,34)(30,36,37,38)(35,39);(10,19)(14,22)(18,23)(20,21)(24,27,28,35)(25,30,31,37)(26,33,34,39)(29,36,32,38)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1044);