Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#13158 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1936 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C22×C7×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
316 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)(8,9,14,19,16,20,15,30)(10,12,18,31,23,21,33,13)(11,24,28,39,29,25,34,22)(17,35,36,27,38,26,37,32);(8,10,21,19,15,18,31,20,16,23,12,30,14,33,13,9)(11,25,26,17,28,39,32,37,29,24,35,38,34,22,27,36);(8,11,12,26,15,28,13,32,16,29,21,35,14,34,31,27)(9,17,10,22,19,37,18,25,20,38,23,39,30,36,33,24)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6146);