Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13291 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#13291 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1993 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C23×C7 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
306 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)(10,21,25,37)(11,27,30,35)(12,31)(13,18)(14,20)(15,16,29,34)(17,22)(19,33)(23,32)(24,28,26,39)(36,38);(8,10,22,35,33,24,12,16,14,25,38,27,13,26,32,34)(9,15,23,39,18,30,36,37,20,29,31,28,19,11,17,21);(8,11,14,30)(9,16)(10,23)(12,28,32,39)(13,29,33,15)(17,26)(18,27)(19,35)(20,34)(21,22,37,38)(24,36)(25,31)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6279);