Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#23197 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | D42 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C2×C7 | #2366 |
| prime radical | C7×#3150 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ G | |
880 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,20,25,36)(11,26)(12,16)(13,17)(14,19)(15,23,33,24)(18,30)(21,38,22,31)(27,34)(28,37,39,32)(29,35);(8,10,13,24,14,25,30,23)(9,15,17,36,19,33,18,20)(11,21,12,28,27,22,29,39)(16,31,34,32,35,38,26,37);(8,11,13,12,14,27,30,29)(9,16,18,26,19,35,17,34)(10,21,24,28,25,22,23,39)(15,31,20,37,33,38,36,32);(8,12,13,27,14,29,30,11)(9,16,17,34,19,35,18,26)(10,22,23,28,25,21,24,39)(15,32,20,31,33,37,36,38)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16185);