Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23906 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#24020 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C7×#3487 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,17)(18,22)(19,24)(20,28)(21,25)(23,26)(27,31)(29,30);(9,12)(13,15)(16,18)(17,22)(19,26)(20,29)(21,27)(23,24)(25,31)(28,30);(8,10,11,14)(9,13,12,15)(16,18)(17,23)(19,26)(20,29)(21,27)(22,28)(24,31)(25,30);(9,12)(13,15)(16,19,21,20)(17,24,25,28)(18,26,27,29)(22,23,31,30)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 17008);