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) = C22 |
| direct product | C7×#13172 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #466 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C23×C7 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
274 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)(8,9,11,10)(13,19)(14,23)(15,17)(16,29)(20,33)(21,37)(22,28)(24,30)(25,39)(26,38)(27,40)(31,35)(36,43)(41,42);(8,10,11,9)(12,13)(14,20)(15,28)(16,21)(17,19)(18,22)(23,35)(24,34)(25,41)(26,31)(27,36)(29,32)(30,37)(33,39)(38,43)(40,42);(12,14,24,38,17,26,16,23,18,27,29,39,15,25,30,40)(13,20,34,43,19,31,21,35,22,36,32,33,28,41,37,42)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6160);