Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#23332 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C24 |
| commutator | C23 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C23×C7 | C23×C4 |
| prime radical | C7×#392 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ G | |
770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(9,12)(10,14)(16,17)(18,20)(19,21)(22,23)(24,25,29,33,28,30,27,32)(26,34,36,39,35,31,37,38);(8,9)(10,15)(11,12)(13,14)(16,18)(17,20)(19,22)(21,23)(25,30)(26,35)(32,33)(36,37);(9,12)(13,15)(17,21)(18,22)(24,26)(25,31)(27,37)(28,35)(29,36)(30,34)(32,39)(33,38);(8,10,11,14)(9,13,12,15)(17,21)(20,23)(24,27,28,29)(25,32,30,33)(26,36,35,37)(31,38,34,39)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16320);