Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#22468 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | D42 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C2×C7×#578 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ G | |
1316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,17,23,19,15,18,22)(14,24,28,25,29,36,27,20)(16,30,33,39,34,21,32,38)(26,37,31,35,43,40,42,41);(12,14)(13,20)(15,25)(16,31)(17,28)(18,27)(19,29)(21,37)(22,36)(23,24)(26,32)(30,40)(33,43)(34,42)(35,38)(39,41);(9,11)(12,15,19,13)(14,25,29,20)(16,30,34,21)(17,23,18,22)(24,27,36,28)(26,41,43,35)(31,40,42,37)(32,39,33,38);(12,16)(13,21)(14,26)(15,30)(17,33)(18,32)(19,34)(20,35)(22,39)(23,38)(24,40)(25,41)(27,42)(28,31)(29,43)(36,37)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 15456);