Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C4 |
| direct product | C7×#13117 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #476 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1920 |
| prime radical | C23×C7 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
254 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)(8,9,11,15,14,19,13,17)(10,20,23,34,24,37,21,36)(12,26,25,35,29,33,28,18)(16,27,31,22,32,38,30,39);(9,15,19,17)(16,30,32,31)(18,26,35,33)(20,36,37,34);(8,10,11,21,14,24,13,23)(9,16,17,31,19,32,15,30)(12,27,25,39,29,38,28,22)(18,34,33,37,35,36,26,20)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6105);