Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#8258 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | C4×C22⋊C4 |
| prime radical | C23×C7 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,14)(13,15)(16,17,20,25,21,26,29,24)(18,19,23,30,28,31,27,22);(8,10,12,15,13,14,11,9)(16,18,20,23,21,28,29,27)(17,22,25,19,26,30,24,31);(8,11,13,12)(9,10,15,14)(16,19,29,22,21,31,20,30)(17,23,24,18,26,27,25,28)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1246);