Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8803 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#8803 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #383 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C23×C7 | #126 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
216 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)(10,18,21,22)(11,14)(12,16)(13,17)(15,20,23,19)(24,25)(26,34,36,39)(27,31,37,30)(28,32)(29,33)(35,38);(9,14,17,16)(15,22,23,18)(24,26,35,37,29,36,28,27)(25,30,32,34,33,31,38,39);(8,10,11,19,13,21,12,20)(9,15,16,18,17,23,14,22)(24,27,28,36,29,37,35,26)(25,31,32,39,33,30,38,34)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1791);