Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#8090 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×C22⋊C4 |
| 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 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C42×C7 ◁ G | |
544 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)(10,17)(11,15)(12,21)(13,16)(14,18)(19,22)(20,23)(24,25,27,31,29,32,28,37)(26,33,35,39,34,30,36,38);(8,10)(9,14,16,23)(11,20)(12,18)(13,19)(15,17,21,22)(24,26,28,36,29,34,27,35)(25,30,31,38,32,33,37,39);(8,11,13,12)(9,15,16,21)(10,18,19,20)(14,22,23,17)(26,34)(30,33)(35,36)(38,39)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1078);