Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#7987 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #95 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | C2×#95 |
| prime radical | C24×C7 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C42×C7 ◁ G | |
542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,11,16,13,17,12,15)(10,18,19,14,21,22,20,23)(24,25,28,33,29,32,27,31)(26,34,35,39,36,30,37,38);(8,10,12,20,13,21,11,19)(9,14,16,22,17,23,15,18)(24,26)(25,30)(27,35)(28,37)(29,36)(31,38)(32,34)(33,39);(24,27,29,28)(25,31,32,33)(26,35,36,37)(30,38,34,39)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 975);