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×#13156 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1936 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
306 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)(9,15,19,26)(10,21,24,29)(11,16)(12,27)(13,30)(17,32)(18,35,28,20)(23,37,33,25)(31,36)(34,38);(8,9,16,18,30,35,11,26,14,19,12,28,13,20,27,15)(10,22,29,38,37,17,33,34,24,39,21,31,25,32,23,36);(8,10,11,23,13,25,16,21,14,24,27,33,30,37,12,29)(9,17,15,31,20,22,28,34,19,32,26,38,35,39,18,36)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6144);