Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#12842 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #466 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C7×#131 ◁ G | |
366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,17,21,18,15,29,22)(14,23,27,38,28,40,43,39)(16,31,24,20,32,36,41,30)(19,33,34,37,25,42,35,26);(12,14,24,33,17,27,32,37,18,28,41,42,29,43,16,26)(13,19,31,39,22,35,30,40,15,25,36,38,21,34,20,23);(9,11)(12,15)(13,18)(14,25)(16,30)(17,21)(19,28)(20,32)(22,29)(23,37)(24,36)(26,40)(27,34)(31,41)(33,38)(35,43)(39,42)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5830);