Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19230 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#19237 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| prime radical | C7×#3245 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,12)(11,13)(14,15)(16,17,21,24)(18,25,27,30)(19,23,20,29)(22,28,31,26);(9,12)(13,14)(16,18,20,26,21,27,19,28)(17,22,23,25,24,31,29,30);(9,13)(11,15)(12,14)(16,17)(18,22)(19,29)(20,23)(21,24)(25,26)(27,31)(28,30);(16,19,21,20)(17,23,24,29)(18,26,27,28)(22,30,31,25)301 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12225);