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) = C4 |
| direct product | C7×#13113 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×D8 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C22×C7 | C4×D8 |
| prime radical | C23×C7 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C7×C8 ◁ C2×C4×C7×C8 ◁ G | |
262 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)(10,13)(11,12)(14,15)(16,17,22,27,20,25,36,28)(18,26,31,40,30,23,21,37)(19,32,34,44,33,47,35,39)(24,41,43,46,42,38,45,29);(9,12)(13,15)(16,18,22,31,20,30,36,21)(17,23,27,37,25,26,28,40)(19,29,34,41,33,46,35,38)(24,39,43,32,42,44,45,47);(8,10,11,14)(9,13,12,15)(16,19,20,33)(17,24,25,42)(18,29,30,46)(21,38,31,41)(22,34,36,35)(23,39,26,44)(27,43,28,45)(32,40,47,37)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6101);