Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#8076 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C7 | C22⋊C4 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | C2×#95 |
| prime radical | C24×C7 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42×C7 ◁ C22×C42×C7 ◁ G | |
564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,18)(11,15)(12,16)(13,17)(14,19)(20,23)(21,22)(24,25,27,30,28,31,35,38)(26,32,34,39,33,29,36,37);(8,10,12,20,13,21,11,19)(9,14,15,22,17,23,16,18)(24,26)(25,29,31,32)(27,34)(28,33)(30,37,38,39)(35,36);(8,11,13,12)(9,15,17,16)(10,19,21,20)(14,22,23,18)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1064);