Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C24 |
| direct product | C7×#12986 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #386 |
| commutator | C2×C4 | C22×C7×C8 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1869 |
| prime radical | C2×C7×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C7×C8 ◁ G | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,14,19,13,18,32,36,15,20,34,38,12,31,22,23)(10,21,25,39,26,11,28,30,27,29,17,37,24,16,35,33);(8,10,22,35,12,24,34,17,15,27,32,28,13,26,14,25)(9,16,36,39,31,29,19,33,20,11,23,37,18,21,38,30);(8,11)(9,17)(10,23)(12,21)(13,29)(14,33)(15,16)(18,35)(19,26)(20,25)(22,37)(24,38)(27,36)(28,31)(30,34)(32,39)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5974);