Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11295 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #124 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C22×C7 | #400 |
| prime radical | C24×C7 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
574 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)(12,13,17,22,18,23,16,21)(14,24,27,19,28,35,26,36)(15,29,31,20,33,38,32,39)(25,41,42,37,43,34,30,40);(8,9)(10,11)(12,14,16,27,18,28,17,26)(13,19,21,35,23,36,22,24)(15,30,32,25,33,42,31,43)(20,37,29,34,39,40,38,41);(9,11)(12,15,16,31,18,33,17,32)(13,20,21,38,23,39,22,29)(14,25,26,42,28,43,27,30)(19,34,24,40,36,41,35,37)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4283);