Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×C7×#1379 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C7×C8 | D4 |
| commutator | C4 | C2×C4×C7×C8 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C23×C7 | C22⋊C8 |
| prime radical | C24×C7 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7×C8 ◁ C23×C7×C8 ◁ G | |
606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2)(39,40,41,42,43,44,45);(7,8,11,16,12,17,26,34,13,18,27,35,10,15,24,32)(9,19,21,36,20,30,37,33,22,31,25,38,23,14,28,29);(3,4,5,6)(7,9)(8,14)(10,23)(11,25)(12,20)(13,22)(15,31)(16,33)(17,19)(18,30)(21,27)(24,37)(26,28)(29,35)(32,36)(34,38)784 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5541);