Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C4 |
| direct product | C7×#9588 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #121 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | #388 |
| prime radical | C7×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(9,13)(10,17)(11,14)(15,23)(16,21)(18,20)(24,25,27,30,29,33,28,32)(26,34,36,39,35,38,37,31);(8,9,14,21,12,16,11,13)(10,18,22,20,17,15,19,23)(25,30,33,32)(26,35)(31,38,39,34)(36,37);(8,10,11,19,12,17,14,22)(9,15,13,20,16,18,21,23)(24,26,27,36,29,35,28,37)(25,31,32,38,33,39,30,34)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 2576);