Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C25 |
| direct product | C11×#537 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C11 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ G | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 352 |
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)(18,22)(19,20)(21,23);(17,20)(21,22);(12,13)(14,15)(16,17)(18,21)(19,20)(22,23);(16,18,19,23)(17,21,20,22)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 1317);