Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C29 |
| direct product | C7×#60182 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C23×D4 |
| commutator | C4 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C2×C7 | C22×C4×D4 |
| prime radical | C2×C7×#575 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C7×M16 ◁ G | |
2628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,18,14,17,12,26)(10,19,21,15,22,30,27,29)(11,16,25,32,23,31,24,36)(20,33,35,28,34,39,37,38);(8,10)(9,15)(11,20)(12,27)(13,21)(14,22)(16,28)(17,29)(18,30)(19,26)(23,34)(24,37)(25,35)(31,38)(32,39)(33,36);(8,11,13,25,14,23,12,24)(9,16,18,32,17,31,26,36)(10,20,21,35,22,34,27,37)(15,28,30,39,29,38,19,33);(11,23)(16,31)(20,34)(24,25)(28,38)(32,36)(33,39)(35,37);(9,17)(15,29)(16,31)(18,26)(19,30)(28,38)(32,36)(33,39)574 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53170);