Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#10804 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #117 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C22×C7 | #382 |
| prime radical | C24×C7 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C7×#117 ◁ G | |
642 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)(15,26,28,19)(16,20)(17,21)(18,24,31,39)(23,36)(27,41,42,32)(29,33)(30,37)(34,38)(40,43);(12,14)(13,18,21,31)(15,23)(16,24)(17,25)(19,30,26,40)(20,22,33,35)(27,37)(28,38)(29,39)(32,34,41,36)(42,43);(9,11)(12,15,16,27,17,28,29,42)(13,19,20,32,21,26,33,41)(14,23,24,37,25,38,39,43)(18,30,22,34,31,40,35,36)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 3792);