Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C27 |
| direct product | C7×#21006 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#121 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C22×C7×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
1654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,17,23,18,22,16,21)(14,19,27,34,26,35,25,33)(15,28,29,20,30,36,31,37)(24,38,39,32,41,42,40,43);(12,14,16,27,18,26,17,25)(13,19,23,33,22,35,21,34)(15,24,29,40,30,41,31,39)(20,32,28,42,37,43,36,38);(9,11)(12,15)(13,20)(14,24)(16,29)(17,31)(18,30)(19,32)(21,36)(22,37)(23,28)(25,39)(26,41)(27,40)(33,42)(34,38)(35,43);(12,16,18,17)(13,21,22,23)(14,25,26,27)(15,29,30,31)(19,33,35,34)(20,36,37,28)(24,39,41,40)(32,42,43,38)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13994);