Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×#364 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C11 | D4 |
| commutator | C4 | C2×C8×C11 |
| Frattini, cosocle | C2×C8 | C22×C11 |
| socle | C2×C11 | C4⋊C8 |
| prime radical | C22×C11 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C11 ◁ C2×C8×C11 ◁ G | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16,21,17,20,25,27,18,22,24,19,15,26,14,23);(12,14,15,24,18,25,17,16)(13,19,20,23,22,21,26,27)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 168);