Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#13151 |
| 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 | |
332 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)(12,13,17,24,19,25,18,34)(14,26,29,37,31,42,30,40)(15,23,32,36,33,41,16,22)(20,35,38,43,39,28,21,27);(12,14,15,27,17,29,32,35,19,31,33,43,18,30,16,28)(13,20,36,42,34,21,23,37,25,39,22,26,24,38,41,40);(9,11)(12,14,15,27,17,29,32,35,19,31,33,43,18,30,16,28)(13,21,22,40,34,39,41,42,25,38,36,37,24,20,23,26)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6139);