Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7×C8 · Schur multiplier (H2) = C29 |
| direct product | C7×#60276 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C7 | C24 |
| commutator | C2 | C24×C7×C8 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C22×C7 | C24×C4 |
| prime radical | C22×C7×#41 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C7 ◁ G | |
2306 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)(8,9,10,12,11,13,14,15)(16,17)(18,24)(19,23)(20,22)(21,25)(26,30)(27,29)(28,31);(16,18)(17,21)(19,26)(20,25)(22,24)(23,28)(27,31)(29,30);(18,25)(21,24)(26,31)(28,30);(17,22)(21,24)(23,29)(28,30);(16,19,20,27)(17,23,22,29)(18,26,25,31)(21,28,24,30)952 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53264);