Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13033 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13036 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1879 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
476 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)(8,9,14,19,21,24,13,18,16,20,32,33,15,10,12,23)(11,25,28,17,34,35,26,22,30,38,31,37,29,39,27,36);(8,10,21,20,16,24,15,9)(11,22,29,37,30,36,34,17)(12,23,14,33,13,18,32,19)(25,28,35,27,38,31,39,26);(8,11)(9,17)(10,22)(12,31)(13,28)(14,26)(15,34)(16,30)(18,35)(19,38)(20,37)(21,29)(23,39)(24,36)(25,33)(27,32)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6024);