Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of 3−1+2⋊Q8 |
| automorphism | Aut(G) = D4×AGL(1,ℤ/9) (order 432) · Inn(G) = C2×AGL(1,ℤ/9) · Out(G) = C22 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | 3−1+2⋊Q8 |
| unambiguous semidirect product | Dic18⋊C3 |
| Sylow | 3−1+2 ⋊ Q8 |
39 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,5,6,7,2,8,3,9,4)(10,12,13,17)(11,14,15,16);(2,4)(3,7)(5,6)(8,9)(10,11,13,15)(12,16,17,14)Polycyclic presentation:
2^2,3,2,3^2|a^2=d,b^2=d,e^3=f^2,b^a=bd,e^a=e^2f,f^a=f^2,e^c=ef31 conjugacy classes — view the full character table.
| maximal subgroup of | Dic36⋊C3, #3664⋊C3, #3672⋊C3, … |
| maximal normal subgroup of | Dic36⋊C3, #3664⋊C3, #3672⋊C3, … |
| maximal quotient of | (3−1+2⋊C4)⋊C4, C4⋊(3−1+2⋊C4), C2×3−1+2⋊Q8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 52);