Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#18087 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C42 | C24×C7 |
| socle | C22×C7 | C2×C4×D4 |
| prime radical | C7×#466 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C7×#431 ◁ G | |
606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,15)(11,17)(12,16)(13,18)(14,19)(20,23)(21,22)(24,25,29,32,30,26,28,33)(27,31,36,38,37,34,35,39);(8,10)(9,14)(11,19)(12,21)(13,20)(15,16)(17,23)(18,22)(24,26)(25,30)(27,34)(28,32)(29,33)(31,37)(35,38)(36,39);(8,10)(9,15)(11,21)(12,19)(13,20)(14,16)(17,22)(18,23)(24,27,29,36,30,37,28,35)(25,31,32,38,26,34,33,39);(8,11,13,12)(9,16,18,17)(10,19,20,21)(14,15,22,23)(24,28,30,29)(25,32,26,33)(27,35,37,36)(31,38,34,39)490 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 11075);