Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C25 |
| direct product | C11×#546 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C11 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ G | |
334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,15)(16,17,20,21)(18,22,23,19);(17,21)(19,22);(12,13)(14,15)(16,18)(17,19)(20,23)(21,22);(16,19)(17,18)(20,22)(21,23)275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 1326);