Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#17323 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7×C8 | C23 |
| commutator | C22 | C22×C42×C7 |
| Frattini, cosocle | C42 | C24×C7 |
| socle | C22×C7 | C22×C42 |
| prime radical | C24×C7 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C4×C7×C8 ◁ G | |
874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,10,13,11,12,14,15)(16,17)(18,21)(19,23)(20,22)(24,28)(25,27)(26,29)(30,31);(16,18,19,24,20,25,26,30)(17,21,23,28,22,27,29,31);(17,22)(21,27)(23,29)(28,31);(9,12)(13,15)784 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 10311);