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×#18084 |
| 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×#3169 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C7×#431 ◁ G | |
652 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)(10,14)(11,13)(12,15)(16,19)(17,22)(18,20)(21,23)(24,25,28,32,29,26,33,31)(27,30,35,38,36,34,39,37);(9,13)(11,19)(14,20)(15,16)(18,23)(21,22)(24,26)(25,29)(27,34)(28,31)(30,36)(32,33)(35,37)(38,39);(8,10)(9,14)(11,18)(12,17)(13,20)(15,22)(16,21)(19,23)(24,27,28,35,29,36,33,39)(25,30,32,38,26,34,31,37);(9,15)(10,17)(13,16)(18,23)(25,26)(30,34)(31,32)(37,38)490 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 11072);