Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#17969 |
| 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×#121 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C7×#431 ◁ G | |
582 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)(9,13)(11,18)(14,20)(15,19)(17,22)(21,23)(24,25,29,32,30,26,28,34)(27,31,37,38,35,33,36,39);(8,9,12,15)(10,14,16,21)(11,19,18,13)(17,23,22,20)(24,26,30,25)(27,33,35,31)(28,32,29,34)(36,38,37,39);(8,10)(9,14)(11,17)(12,16)(13,20)(15,21)(18,22)(19,23)(24,27,29,37,30,35,28,36)(25,31,32,38,26,33,34,39);(10,16)(14,21)(17,22)(20,23)(25,26)(27,35)(32,34)(36,37)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 10957);