Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C24 |
| direct product | C2×C17×#357 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C17 | D8 |
| commutator | C8 | C22×C4×C17 |
| Frattini, cosocle | C2×C8 | C23×C17 |
| prime radical | C22×D8×C17 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C17 ◁ C22×C4×C17 ◁ C22×C8×C17 ◁ G | |
272 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39);(3,4,5,6)(7,8)(9,12)(10,19)(11,15)(13,16)(14,17)(18,21)(20,22);(8,12)(9,16)(10,17)(13,14)(15,21)(18,22)(19,20)748 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 868);