Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#12667 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | #117 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1770 |
| prime radical | C23×C7 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C2×C7×M32 ◁ G | |
296 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)(8,9,13,18,14,19,28,34,12,17,25,33,27,11,16,26)(10,15,21,30,22,31,36,39,23,20,29,35,37,24,32,38);(8,10)(9,15,17,20)(11,24,19,31)(12,23)(13,21)(14,22)(16,32)(18,30,33,35)(25,29)(26,38,34,39)(27,37)(28,36);(8,11,25,34,14,9,16,33,12,19,13,26,27,17,28,18)(10,20,21,35,22,24,36,38,23,15,29,30,37,31,32,39)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5655);