Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#18067 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C42 | C24×C7 |
| socle | C22×C7 | C2×C4×D4 |
| prime radical | C7×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C4×C7×M16 ◁ G | |
654 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)(9,12)(10,14)(13,15)(16,17,21,24,20,23,29,31)(18,25,27,28,26,22,30,19);(8,9,11,13)(10,15,14,12)(16,18)(17,22)(19,24)(20,26)(21,27)(23,25)(28,31)(29,30);(16,19,21,25,20,28,29,22)(17,18,24,27,23,26,31,30);(18,26)(19,28)(22,25)(27,30)490 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 11055);