Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#13201 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #390 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1955 |
| prime radical | C7×#525 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C7×C8 ◁ G | |
420 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)(8,9,13,19,14,18,33,37,15,20,34,23,12,17,22,28)(10,21,25,39,24,31,36,38,26,29,16,35,32,11,27,30);(8,10,22,36,12,32,34,25,15,26,33,27,14,24,13,16)(9,11,28,39,17,29,23,30,20,31,37,35,18,21,19,38);(8,11,12,29,15,31,14,21)(9,16,18,27,20,25,17,36)(10,23,24,28,26,19,32,37)(13,30,22,35,34,38,33,39)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6189);