Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7217 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#7217 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #340 |
| commutator | C2×C8 | C42×C7 |
| Frattini, cosocle | C2×C4×C8 | C22×C7 |
| prime radical | C23×C7 | #102 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ C2×C4×C7×C8 ◁ G | |
226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,16,18,22)(11,14)(12,15)(13,17,20,23)(19,21)(25,30)(26,34)(27,36)(28,35)(31,32)(33,39)(37,38);(8,10)(9,13,15,20)(11,17)(12,18)(14,16,21,22)(19,23)(24,25,27,31,28,32,26,30,29,33,34,38,35,37,36,39)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 205);