Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C2×C8 |
| direct product | C7×#8158 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | C4×C22⋊C4 |
| prime radical | C24×C7 | C42 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7)(8,9)(10,13)(11,15)(12,14)(16,20)(17,19)(18,21)(22,23)(24,25,28,32,29,33,40,46)(26,34,37,48,36,30,41,43)(27,31,38,44,39,45,51,54)(35,47,50,55,49,42,52,53);(8,10,11,16,12,17,18,22)(9,13,15,20,14,19,21,23)(24,26)(25,30,33,34)(27,35)(28,37)(29,36)(31,42,45,47)(32,43,46,48)(38,50)(39,49)(40,41)(44,53,54,55)(51,52);(9,14)(13,19)(15,21)(20,23)(24,27,28,38,29,39,40,51)(25,31,32,44,33,45,46,54)(26,35,37,50,36,49,41,52)(30,42,43,53,34,47,48,55)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1146);