Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#13386 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C7×#3054 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C22×C7×M16 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(9,10)(12,13)(14,17)(15,24)(16,19)(18,22)(20,23)(21,25)(26,27);(8,9)(10,11)(14,21)(17,25)(18,26)(22,27);(8,10,11,9)(13,17)(14,22)(15,23)(18,20)(19,25)(21,27)(24,26)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6374);