Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C22×C4 |
| direct product | C11×#410 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | C22⋊C4 |
| commutator | C22 | C22×C4×C11 |
| Frattini, cosocle | C23 | C23×C11 |
| socle | C2×C11 | C2×C22⋊C4 |
| prime radical | C2×C11×#41 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C22×C4×C11 ◁ G | |
158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,20,22,17)(15,18)(16,19)(21,26,27,24)(23,25);(12,14)(13,17)(15,21)(16,22)(18,24)(19,20)(23,27)(25,26);(12,15,16,23)(13,18,19,25)(14,21,22,27)(17,24,20,26)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 816);