Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#12868 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C2×D8 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C24×C7 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C23×C4×C7 ◁ C23×C7×C8 ◁ G | |
426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3,4,5,6,7)(8,9,10,11)(12,13)(14,17)(15,16)(18,19)(21,25)(22,28)(24,31)(26,34)(27,30)(29,35)(32,33);(13,16)(17,19)(20,21,23,25)(22,26,29,33)(24,27,30,31)(28,32,35,34);(12,14,15,18)(13,17,16,19)(20,22,23,29)(21,26,25,33)(24,32,30,34)(27,35,31,28)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5856);