Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#60218 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C24 |
| commutator | C22 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C22×C7 | C24×C4 |
| prime radical | C7×#3480 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ G | |
1950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,17,20,25,21,24,29,31)(18,22,27,30,26,19,23,28);(9,12)(13,15)(16,18)(17,22)(19,24)(20,27)(21,26)(23,29)(25,30)(28,31);(16,19,20,28,21,22,29,30)(17,23,25,18,24,27,31,26);(8,10,11,14)(9,13,12,15)(16,18)(17,22)(19,24)(20,27)(21,26)(23,29)(25,30)(28,31);(17,24)(18,26)(23,27)(25,31)700 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53206);