Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#12624 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C2×#99 |
| commutator | C23 | C2×C42×C7 |
| Frattini, cosocle | C2×#41 | C23×C7 |
| socle | C2×C7 | C4×#99 |
| prime radical | C7×#525 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C7×C8 ◁ C2×C7×#131 ◁ G | |
456 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)(8,9,13,11,14,17,28,24)(10,18,21,33,22,31,35,36)(12,16,26,23,27,32,39,37)(15,29,19,25,30,38,34,20);(8,10)(9,15)(11,19)(12,25,27,20)(13,21)(14,22)(16,31,32,18)(17,30)(23,36,37,33)(24,34)(26,38,39,29)(28,35);(8,11,13,17,14,24,28,9)(10,19,21,30,22,34,35,15)(12,23,26,32,27,37,39,16)(18,20,33,29,31,25,36,38)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5612);