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×#543 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C11 |
| prime radical | C24×C11 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ G | |
242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16,19)(14,20,22,18)(15,17,23,25)(21,26,27,24);(13,17)(14,21)(19,25)(22,27);(12,14,16,22)(13,18,19,20)(15,21,23,27)(17,24,25,26);(13,17)(18,24)(19,25)(20,26)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 1323);