Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C11 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×C11×C22⋊C4 |
| unambiguous semidirect product | C22⋊(C42×C11) |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C11 | C22 |
| commutator | C2 | C2×C42×C11 |
| Frattini, cosocle | C23 | C23×C11 |
| socle | C23×C11 | C23 |
| prime radical | C24×C11 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C11 ◁ G | |
258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4)(13,14,15,16,17,18,19,20,21,22,23);(5,6,8,10)(7,9,11,12);(6,9)(10,12)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 783);