Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#441 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C11 | C2×D4 |
| commutator | C4 | C22×C4×C11 |
| Frattini, cosocle | C2×C4 | C23×C11 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C42×C11 ◁ G | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,14)(16,17)(18,24)(19,27)(20,22)(21,25)(23,29)(26,31)(28,30);(16,18,20,26)(17,21,22,30)(19,28,29,25)(23,31,27,24);(12,14,15,13)(17,22)(21,30)(23,27)(24,31)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 847);