Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#8246 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | C4×C22⋊C4 |
| prime radical | C23×C7 | #117 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,16,13,17,11,14)(10,15,19,22,20,23,18,21)(24,25,27,30)(26,32,33,29)(28,36,35,31)(34,37,39,38);(9,14,17,16)(15,21,23,22)(24,26,28,34,27,33,35,39)(25,29,31,37,30,32,36,38);(8,10,12,19,13,20,11,18)(9,15,16,22,17,23,14,21)(26,33)(29,32)(34,39)(37,38)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1234);