Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7529 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#7530 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1408 |
| commutator | C4×C8 | C2×C4×C7 |
| Frattini, cosocle | #493 | C22×C7 |
| prime radical | C7×#493 | C22 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C7 ◁ C2×C4×C7 ◁ C42×C7 ◁ C7×#493 ◁ G | |
152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,18,20,35)(11,22,25,32)(12,17)(13,16,24,34)(14,19,28,38)(15,26,31,37)(21,36,27,30)(23,33)(29,39);(8,10,12,21)(9,14,17,28)(11,23,26,39)(13,27,24,20)(15,30,32,35)(16,33,34,29)(18,22,36,31)(19,37,38,25)175 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 518);