Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#8450 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | #120 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C22×C7×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G | |
848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,16,18,15,14,19,17)(21,22)(23,24)(26,27);(9,11)(16,19)(17,18)(20,21,25,27)(22,23,26,24);(12,14)(13,15)(16,17)(18,19)(20,22)(21,23)(24,27)(25,26)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1438);