Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#20528 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C24 |
| commutator | C22 | C22×C42×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C23×C7 | C23×C4 |
| prime radical | C25×C7 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ G | |
1250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3,4,5,6,7)(12,13)(14,18)(15,17)(16,19)(20,21,23,26,24,27,30,33)(22,25,29,32,28,31,34,35);(8,9,10,11)(12,14)(13,16)(15,19)(17,18)(20,22)(21,25)(23,29)(24,28)(26,32)(27,31)(30,34)(33,35);(14,19)(16,18)(22,28)(25,31)(29,34)(32,35);(13,17)(16,18)(22,28)(25,31)(29,34)(32,35)700 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13516);