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×#18088 |
| 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)(8,9,14,19)(10,20,11,16)(12,18,13,17)(15,22,23,21)(24,25,29,32,30,26,28,34)(27,31,36,38,37,33,35,39);(8,10,14,11)(9,15,19,23)(12,21,13,22)(16,17,20,18)(24,26,30,25)(27,33,37,31)(28,32,29,34)(35,38,36,39);(8,11,14,10)(9,16,19,20)(12,21,13,22)(15,17,23,18)(24,27,29,36,30,37,28,35)(25,31,32,38,26,33,34,39);(8,12,14,13)(9,17,19,18)(10,21,11,22)(15,20,23,16)(24,28,30,29)(25,32,26,34)(27,35,37,36)(31,38,33,39)490 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 11076);