Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#10799 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×C42 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C22×C7 | #376 |
| prime radical | C24×C7 | #41 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ G | |
650 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)(14,22,25,35)(15,19)(16,20)(17,21)(18,24,30,38)(23,34,37,42)(26,31)(27,32)(28,33)(29,36,40,43)(39,41);(12,14)(13,18,21,30)(15,23)(16,24)(17,25)(19,29,32,40)(20,22,33,35)(26,36)(27,37)(28,38)(31,34,41,42)(39,43);(9,11)(12,15,16,26,17,27,28,39)(13,19,20,31,21,32,33,41)(14,23,24,36,25,37,38,43)(18,29,22,34,30,40,35,42)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 3787);