Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#13470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C22×C7×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
326 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)(12,13,18,35,19,25,33,24)(14,26,31,38,32,42,43,40)(15,21,27,22,16,23,17,34)(20,36,41,28,37,30,39,29);(8,9)(10,11)(12,14,27,36,18,31,16,28,19,32,17,30,33,43,15,29)(13,20,21,40,24,39,34,42,25,37,23,38,35,41,22,26);(9,11)(13,21)(14,28)(15,27)(16,17)(18,33)(20,37)(22,24)(23,25)(26,38)(29,32)(30,43)(31,36)(34,35)(40,42)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6458);