Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8412 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#8431 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #379 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C7×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 47 points:
(1,2,3,4,5,6,7)(8,9,12,14,13,10,11,15)(16,17,21,27,22,25,20,26)(18,28,30,41,31,23,36,38)(19,32,34,24,35,40,33,39)(29,43,44,37,45,47,46,42);(8,10)(9,13)(11,15)(12,14)(16,18)(17,23)(19,29)(20,30)(21,36)(22,31)(24,37)(25,28)(26,38)(27,41)(32,47)(33,44)(34,46)(35,45)(39,42)(40,43);(8,11,13,12)(9,14,10,15)(16,19,22,35)(17,24,25,39)(18,29,31,45)(20,34,21,33)(23,37,28,42)(26,40,27,32)(30,46,36,44)(38,43,41,47)301 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1419);