Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#12887 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#106 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C23×C7 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
286 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)(18,28,32,43)(19,34,33,47)(20,36)(21,37)(22,27)(23,31,41,44)(24,35,42,45)(25,30)(26,38)(29,40)(39,46);(9,12)(13,15)(16,18,29,44,21,33,46,35,22,32,20,31,38,19,30,45)(17,23,39,43,26,24,40,47,27,41,25,28,37,42,36,34);(8,10,11,14)(9,13,12,15)(16,19,20,35,21,18,30,31,22,33,29,45,38,32,46,44)(17,24,25,34,26,41,36,43,27,42,39,47,37,23,40,28)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5875);