Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#13188 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #476 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1948 |
| prime radical | C23×C7 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,20,15,21,14,19)(10,22,26,36,27,33,25,39)(11,28,29,17,30,18,12,31)(16,32,34,24,35,38,37,23);(8,10)(9,16,21,35)(11,23)(12,32)(13,25)(14,26)(15,27)(17,33,31,22)(18,36,28,39)(19,34,20,37)(24,30)(29,38);(8,11,13,29,15,30,14,12)(9,17,19,28,21,31,20,18)(10,23,25,38,27,24,26,32)(16,33,34,39,35,22,37,36)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6176);