Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#436 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C23 |
| commutator | C22 | C22×C4×C11 |
| Frattini, cosocle | C2×C4 | C23×C11 |
| socle | C22×C11 | C22×C4 |
| prime radical | C2×D4×C11 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ G | |
122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,15)(16,17,20,18,19,21,23,22);(13,15)(17,21)(18,22);(16,18,20,21,19,22,23,17)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 842);