Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C23 |
| direct product | C2×C11×D16 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | D8 |
| commutator, Frattini, cosocle | C8 | C23×C11 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C2×C4×C11 ◁ C2×C8×C11 ◁ G | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29);(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18);(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 911);