Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11305 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #124 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C22×C7 | #397 |
| prime radical | C24×C7 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(8,9,10,11)(12,13)(14,23)(15,20)(16,31)(17,22)(18,21)(19,25)(24,39)(26,40)(27,34)(28,33)(29,37)(30,38)(32,36)(35,41)(42,43);(12,14,18,27,17,26,16,25)(13,19,31,40,22,34,21,23)(15,28,29,41,30,42,32,24)(20,35,36,33,38,39,37,43);(12,15,16,29,17,30,18,32)(13,20,21,36,22,38,31,37)(14,24,25,28,26,41,27,42)(19,33,23,39,34,43,40,35)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4293);