Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C22 |
| direct product | C7×#12011 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | C2×D8 |
| commutator | C8 | C22×C7×C8 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C2×C7 | C2×#335 |
| prime radical | C23×C7 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C22×C7×C8 ◁ C2×C4×C7×C8 ◁ G | |
220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,17,13,19,11,27,14,20,22,16,28,18,29,34)(10,21,25,32,26,36,24,15,23,31,37,33,39,35,38,30);(8,10,22,37,13,26,29,38,14,23,12,25,28,39,11,24)(9,15,27,35,18,32,17,31,20,30,34,36,19,33,16,21);(10,23)(15,30)(21,31)(24,38)(25,37)(26,39)(32,33)(35,36)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4999);