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×C8×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C24 |
| commutator | C2 | C24×C7×C8 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C22×C7 | C24×C4 |
| prime radical | C2×C7×2−1+4 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ G | |
2190 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)(8,9,10,11,12,13,14,15);(17,21)(19,27)(22,24)(25,31);(16,17,20,21)(18,24,26,22)(19,23,27,28)(25,30,31,29);(16,18,20,26)(17,22,21,24)(19,25,27,31)(23,29,28,30);(16,19)(17,23)(18,25)(20,27)(21,28)(22,29)(24,30)(26,31)952 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53317);