Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#8189 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C7 | C2×D4 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | C4×C22⋊C4 |
| prime radical | C2×C7×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,16,13,17,11,14)(10,15,19,22,20,23,18,21)(24,25)(26,29)(27,31)(28,30)(32,36)(33,35)(34,37)(38,39);(9,14,17,16)(15,21,23,22)(24,26,27,33,28,32,34,38)(25,29,31,35,30,36,37,39);(8,10,12,19,13,20,11,18)(9,15,16,22,17,23,14,21)(25,30)(26,32)(31,37)(33,38)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1177);