Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#60319 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C25 |
| commutator | C22 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C22×C7 | C24×C4 |
| prime radical | C7×#3460 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ G | |
1848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,14)(11,13)(12,15)(16,17,21,20,22,25,26,30)(18,23,19,24,27,29,28,31);(8,10)(9,12)(11,15)(13,14)(16,18)(17,23)(19,21)(20,24)(22,27)(25,29)(26,28)(30,31);(10,15)(12,14)(16,19)(17,24)(18,26)(20,29)(21,27)(22,28)(23,30)(25,31);(9,13)(12,14)(18,27)(19,28)(23,29)(24,31);(16,20,21,25,22,30,26,17)(18,24,19,29,27,31,28,23)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53307);