Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#12008 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C7 | D8 |
| commutator | C8 | C22×C7×C8 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C22×C7 | C2×#107 |
| prime radical | C23×C7 | #107 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C7 ◁ C2×C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
242 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)(8,9,10,12,11,13,14,15)(16,17)(18,27)(19,24)(20,34)(21,36)(22,26)(23,30)(25,28)(29,40)(31,46)(32,42)(33,39)(35,47)(37,44)(38,45)(41,43);(16,18,28,43,21,31,35,45,22,32,33,44,19,29,20,30)(17,23,34,40,24,37,39,42,26,38,47,46,36,41,25,27);(16,19,22,21)(17,24,26,36)(18,29,32,31)(20,33,35,28)(23,37,38,41)(25,34,39,47)(27,40,42,46)(30,44,45,43)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4996);