Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#12670 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | D8 |
| commutator | C8 | C2×C42×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C22×C7 | C4×D8 |
| prime radical | C23×C7 | #135 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ C22×C7×C8 ◁ C2×C4×C7×C8 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(12,13,17,23,18,24,16,22,19,25,33,41,32,40,15,21)(14,26,29,20,30,37,34,38,31,42,27,39,35,36,28,43);(8,9,10,11)(12,14)(13,20)(15,27)(16,29)(17,34)(18,35)(19,31)(21,36)(22,37)(23,42)(24,43)(25,39)(26,41)(28,33)(30,32)(38,40);(12,15,32,33,19,16,18,17)(13,21,40,41,25,22,24,23)(14,27,30,28,31,29,35,34)(20,36,38,26,39,37,43,42)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5658);