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×#17835 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C42 | C24×C7 |
| socle | C22×C7 | C22×C4⋊C4 |
| prime radical | C2×C7×#41 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C4×C7×#41 ◁ G | |
612 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)(8,9,14,11)(10,17,21,23)(12,16,13,22)(15,19,18,20)(24,25,28,32,29,33,27,31)(26,34,36,39,37,38,35,30);(8,10,13,20,14,21,12,19)(9,15,16,17,11,18,22,23)(24,26,27,35,29,37,28,36)(25,30,32,34,33,39,31,38);(8,11,14,9)(10,18,21,15)(12,16,13,22)(17,20,23,19)(24,27,29,28)(25,31,33,32)(26,35,37,36)(30,38,39,34);(8,12,14,13)(9,16,11,22)(10,19,21,20)(15,17,18,23)(24,27,29,28)(25,31,33,32)(26,35,37,36)(30,38,39,34)490 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 10823);