Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#11950 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #102 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C22×C7 | C2×#102 |
| prime radical | C23×C7 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7)(8,9,12,17,13,16,11,15)(10,18,19,23,20,14,21,22)(24,25,28,33,30,35,43,32,29,34,44,51,42,50,27,41)(26,36,38,47,40,49,54,53,39,52,55,31,45,46,37,48);(8,10)(9,14)(11,19)(12,21)(13,20)(15,22)(16,18)(17,23)(24,26,29,39)(25,31,34,47)(27,38,43,55)(28,37,44,54)(30,45,42,40)(32,49,41,46)(33,52,51,36)(35,53,50,48);(8,11,13,12)(9,15,16,17)(10,19,20,21)(14,22,18,23)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4938);