Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13033 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13034 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1879 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 71 points:
(1,2,3,4,5,6,7)(8,9,13,18,14,20,39,51,15,21,12,19,35,50,36,54)(10,22,25,43,26,48,37,44,27,58,24,55,40,52,33,16)(11,28,30,56,31,49,34,17,32,45,29,46,41,53,38,47)(23,42,60,66,61,67,63,70,62,68,59,71,65,69,64,57);(8,10)(9,16,21,44)(11,23)(12,33)(13,37)(14,40)(15,27)(17,42,47,68)(18,48,19,52)(20,55,50,43)(22,54,58,51)(24,36)(25,39)(26,35)(28,57,45,70)(29,64)(30,63)(31,65)(32,62)(34,60)(38,59)(41,61)(46,69,56,67)(49,71,53,66);(8,11,15,32)(9,17,21,47)(10,23,27,62)(12,34,13,38)(14,41,35,31)(16,42,44,68)(18,49,19,53)(20,56,50,46)(22,57,58,70)(24,63,25,64)(26,65,40,61)(28,54,45,51)(29,36,30,39)(33,60,37,59)(43,69,55,67)(48,71,52,66)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6022);