Finite Group DatabasebetaFinite Group Databasebeta| order | 2176 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×D4⋊C16 |
| unambiguous semidirect product | D4⋊(C16×C17) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C17 | D4 |
| commutator | C4 | C2×C16×C17 |
| Frattini, cosocle | C4×C8 | C22×C17 |
| socle | C22×C17 | C22⋊C8 |
| prime radical | C2×D4×C17 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C17 ◁ C4×C8×C17 ◁ G | |
110 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 110 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,22)(21,24)(23,25)(26,27,28,31,29,32,34,37,30,33,35,38,36,39,40,41);(19,22)(20,25)(23,24)952 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2176, 61);