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) = C4 |
| direct product | C7×#7531 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1408 |
| commutator | C4×C8 | C2×C4×C7 |
| Frattini, cosocle | #494 | C22×C7 |
| prime radical | C7×#493 | C4 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C7 ◁ C2×C4×C7 ◁ C42×C7 ◁ C7×#494 ◁ G | |
146 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)(8,9,11,17,14,19,27,34)(10,16,21,32,23,30,37,39)(12,18,26,35,13,28,25,33)(15,24,29,22,31,36,38,20);(9,15)(10,20)(12,13)(16,18)(17,32)(19,31)(21,24)(22,23)(25,26)(28,30)(29,33)(34,39)(35,38)(36,37)175 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 519);