Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C7 · Schur multiplier (H2) = C27 |
| direct product | C7×#20343 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator | C4 | C22×C42×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C22×C7 | C22×C22⋊C4 |
| prime radical | C22×C7×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C7×#41 ◁ G | |
2130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7)(8,9,10,11)(12,13)(14,18)(15,17)(16,20)(19,23)(21,25)(22,24)(26,27);(13,17,20,19)(18,24,25,26);(12,14)(13,18)(15,22)(16,21)(17,24)(19,26)(20,25)(23,27);(14,21)(18,25)(22,27)(24,26)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13331);