Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13175 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#13175 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1943 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C23×C7 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
256 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)(9,15)(10,19,22,35)(11,13)(12,26)(16,30,32,33)(17,27)(18,24)(20,31,37,39)(21,36,23,34)(25,28);(8,9)(10,20,22,37)(11,24)(12,27)(13,15)(14,18)(16,23,32,21)(17,28)(19,33,35,30)(25,29)(26,38)(31,34,39,36);(8,10,12,19,13,23,26,36,14,22,28,35,11,21,25,34)(9,16,17,31,15,20,29,30,18,32,27,39,24,37,38,33)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6163);