Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#13152 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×D8 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C22×C7 | C4×D8 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C7×C8 ◁ C2×C4×C7×C8 ◁ G | |
324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,18,24,19,26,37,25)(14,27,31,40,15,32,35,43)(16,36,38,22,28,23,17,39)(20,34,21,33,42,30,41,29);(12,14,28,33,37,35,38,34,19,15,16,29,18,31,17,30)(13,20,22,27,24,21,23,40,26,42,39,32,25,41,36,43);(9,11)(12,15,16,33,37,31,17,34,19,14,28,29,18,35,38,30)(13,21,22,40,24,42,23,32,26,41,39,43,25,20,36,27)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6140);