Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#13466 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #2065 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,16,21)(10,22)(11,28)(12,19,33,39)(13,18,14,36)(15,23,35,20)(17,30)(24,32)(25,38)(26,29)(27,37)(31,34);(8,10,12,24,15,26,13,34,16,27,33,28,35,17,14,25)(9,11,18,22,20,31,19,30,21,32,36,37,23,38,39,29);(8,11)(9,17)(10,23)(12,30)(13,22)(14,37)(15,38)(16,32)(18,24)(19,25)(20,27)(21,26)(28,36)(29,33)(31,35)(34,39)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6454);