Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#11939 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #102 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C22×C7 | C2×#102 |
| prime radical | C23×C7 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 47 points:
(1,2,3,4,5,6,7)(9,12)(10,14)(13,15)(16,17,21,26,19,24,34,25,22,27,37,42,35,43,20,36)(18,28,31,40,33,44,47,38,32,45,46,23,29,39,30,41);(8,9,11,13)(10,15,14,12)(16,18,22,32)(17,23,27,40)(19,29,35,33)(20,31,34,46)(21,30,37,47)(24,38,43,41)(25,44,36,39)(26,45,42,28);(16,19,22,35)(17,24,27,43)(18,29,32,33)(20,21,34,37)(23,38,40,41)(25,42,36,26)(28,39,45,44)(30,46,47,31)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4927);