Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#541 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C11 |
| prime radical | C23×C11 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ G | |
192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(14,18)(16,19)(20,21,24,27)(22,25,30,28)(23,31,32,26)(29,34,35,33);(12,13,15,17)(14,16,18,19)(20,21)(22,28)(23,26)(24,27)(25,30)(29,34)(31,32)(33,35);(12,14)(13,16)(15,18)(17,19)(20,22)(21,25)(23,29)(24,30)(26,33)(27,28)(31,34)(32,35);(20,23,24,32)(21,26,27,31)(22,29,30,35)(25,33,28,34)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 1321);