Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C11×C16 · Schur multiplier (H2) = C22 |
| direct product | C11×C22⋊C16 |
| unambiguous semidirect product | C22⋊(C11×C16) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C11 | C22 |
| commutator | C2 | C2×C11×C16 |
| Frattini, cosocle | C2×C8 | C22×C11 |
| socle | C22×C11 | C2×C8 |
| prime radical | C23×C11 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C11 ◁ G | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,15)(16,17,18,21,19,22,24,27,20,23,25,28,26,29,30,31);(13,15)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 152);