Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#8265 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C7 | C23 |
| commutator | C23 | C2×C42×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle, prime radical | C23×C7 | C2×C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C7 ◁ G | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 1792 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(8,9,10,11)(12,13,16,18,15,17,19,14)(20,21,23,22,24,25,27,26);(9,11)(13,17)(14,18)(20,21,23,22,24,25,27,26);(12,14,19,17,15,18,16,13)(20,22,23,25,24,26,27,21)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1253);