Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C7 · Schur multiplier (H2) = C24×C4 |
| direct product | C7×#20427 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C24 |
| commutator | C22 | C22×C42×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C23×C7 | C23×C4 |
| prime radical | C24×C7 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ G | |
1118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,10,11)(12,13,15,14)(16,17)(18,21)(19,23)(20,22)(24,27)(25,28)(26,29)(30,31);(9,11)(16,18,19,25,20,24,26,30)(17,21,23,28,22,27,29,31);(12,14,15,13)(18,24)(21,27)(25,30)(28,31);(17,22)(21,27)(23,29)(28,31)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13415);