Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#11473 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#99 |
| commutator | C23 | C22×C7×C8 |
| Frattini, cosocle | C2×C22⋊C4 | C23×C7 |
| prime radical | C2×C7×#120 | C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G | |
548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,11,14,12,15,10,13)(16,17)(18,24)(19,27,28,22)(20,23)(21,26,29,31)(25,30);(16,18)(17,21)(19,26)(20,25)(22,24)(23,29)(27,30)(28,31);(8,10,12,11)(9,13,15,14)(18,25)(21,29)(24,30)(26,31)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4461);