Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C22 |
| direct product | C4×C11×SD16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | D4 |
| commutator | C4 | C22×C4×C11 |
| Frattini, cosocle | C2×C4 | C23×C11 |
| socle | C22×C11 | C2×D4 |
| prime radical | C2×D4×C11 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ C42×C11 ◁ G | |
122 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4)(13,14,15,16,17,18,19,20,21,22,23);(5,6,7,8,9,10,11,12);(6,8)(7,11)(10,12)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 844);