Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43×C7 · Schur multiplier (H2) = C22×C43 |
| direct product | C7×#20331 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C24 |
| commutator | C2 | C2×C43×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C23×C7 | C23×C4 |
| prime radical | C24×C7 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ G | |
1470 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)(16,17,19,23,20,22,26,29)(18,21,25,28,24,27,30,31);(8,9,10,11)(16,18)(17,21)(19,25)(20,24)(22,27)(23,28)(26,30)(29,31);(12,13,14,15)(17,22)(21,27)(23,29)(28,31);(18,24)(21,27)(25,30)(28,31)952 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13319);