Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C7 · Schur multiplier (H2) = C27 |
| direct product | C7×#20354 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C22×C22⋊C4 |
| commutator | C22 | C22×C42×C7 |
| Frattini, cosocle | C24 | C24×C7 |
| prime radical | C23×C7×D4 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C7×D4 ◁ G | |
2410 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)(12,13)(14,20,22,17)(15,18)(16,19)(21,26,27,24)(23,25);(8,9,10,11)(12,14)(13,17)(15,21)(16,22)(18,24)(19,20)(23,27)(25,26);(12,15)(13,18)(14,21)(16,23)(17,24)(19,25)(20,26)(22,27);(15,23)(18,25)(21,27)(24,26)574 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13342);