Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C24×C4 |
| direct product | C7×#25924 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C7×#3038 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
1138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3,4,5,6,7)(8,9,12,15)(10,14,11,13)(16,17)(18,21)(19,22)(20,23);(8,10,12,11)(9,13,15,14)(16,18,21,17)(19,20,23,22);(8,10,12,11)(9,14,15,13)(16,19,20,18,21,23,22,17);(8,11,12,10)(9,14,15,13)(16,20,21,22)(17,19,18,23)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 18912);