Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#358 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C11 | D8 |
| commutator | C8 | C2×C4×C11 |
| Frattini, cosocle | C2×C8 | C22×C11 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C2×C4×C11 ◁ C2×C8×C11 ◁ G | |
58 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15)(17,21)(18,24)(19,26)(22,31)(23,30)(25,28)(27,29);(16,17,20,23)(18,25,27,31)(19,28,26,22)(21,29,30,24)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 162);