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×#542 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C11 |
| prime radical | C11×#121 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ G | |
216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,18)(15,17)(16,19)(22,26)(24,27);(12,13,15,17)(14,16,19,18)(20,21,23,25)(22,24,26,27);(12,13)(14,16)(15,17)(18,19)(20,22)(21,24)(23,26)(25,27);(12,14,15,19)(13,16,17,18)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 1322);