Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C7×C8 · Schur multiplier (H2) = C22 |
| direct product | C7×#7168 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #95 |
| commutator | C2×C4 | C4×C7×C8 |
| Frattini, cosocle | C2×C4×C8 | C22×C7 |
| socle | C22×C7 | #336 |
| prime radical | C23×C7 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3,4,5,6,7)(9,13)(10,11)(14,15)(16,17,20,24,21,25,30,27,22,26,31,18,19,23,29,28);(8,9,11,14,12,15,10,13)(16,18,20,25,21,28,30,26,22,24,31,23,19,27,29,17)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 156);