Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#446 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C23 |
| commutator | C22 | C22×C4×C11 |
| Frattini, cosocle | C2×C4 | C23×C11 |
| socle, prime radical | C22×C11 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ G | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,19)(15,17)(16,18)(20,21,23,22,24,25,27,26);(12,14,16,15)(13,17,18,19);(12,15,16,14)(13,17,18,19)(20,22,23,25,24,26,27,21)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 852);