Group 16.7

Order 16 Exponent 4 Cyclic no Abelian no Elementary no p-Group yes (2) 2-class 2 Rank 2 Nilpotent yes Supersoluble yes Soluble yes Simple no Perfect no

Permutation Representation

Cayley Table:

Elements:

• 1 = ()
• 2 = (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)
• 3 = (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)
• 4 = (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,16)(14,15)
• 5 = (1,5,3,7)(2,6,4,8)(9,13,11,15)(10,14,12,16)
• 6 = (1,6,3,8)(2,5,4,7)(9,14,11,16)(10,13,12,15)
• 7 = (1,7,3,5)(2,8,4,6)(9,15,11,13)(10,16,12,14)
• 8 = (1,8,3,6)(2,7,4,5)(9,16,11,14)(10,15,12,13)
• 9 = (1,9,2,10)(3,11,4,12)(5,15,6,16)(7,13,8,14)
• 10 = (1,10,2,9)(3,12,4,11)(5,16,6,15)(7,14,8,13)
• 11 = (1,11,2,12)(3,9,4,10)(5,13,6,14)(7,15,8,16)
• 12 = (1,12,2,11)(3,10,4,9)(5,14,6,13)(7,16,8,15)
• 13 = (1,13,2,14)(3,15,4,16)(5,9,6,10)(7,11,8,12)
• 14 = (1,14,2,13)(3,16,4,15)(5,10,6,9)(7,12,8,11)
• 15 = (1,15,2,16)(3,13,4,14)(5,11,6,12)(7,9,8,10)
• 16 = (1,16,2,15)(3,14,4,13)(5,12,6,11)(7,10,8,9)

Derived Series

Lower Central Series

Upper Central Series

Composition Series

Chief Series

Maximal Subgroups

Conjugacy Classes

There are 10 conjugacy classes.

{()}

{(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)}

{(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)}

{(1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,16)(14,15)}

{(1,5,3,7)(2,6,4,8)(9,13,11,15)(10,14,12,16), (1,7,3,5)(2,8,4,6)(9,15,11,13)(10,16,12,14)}

{(1,6,3,8)(2,5,4,7)(9,14,11,16)(10,13,12,15), (1,8,3,6)(2,7,4,5)(9,16,11,14)(10,15,12,13)}

{(1,9,2,10)(3,11,4,12)(5,15,6,16)(7,13,8,14), (1,11,2,12)(3,9,4,10)(5,13,6,14)(7,15,8,16)}

{(1,10,2,9)(3,12,4,11)(5,16,6,15)(7,14,8,13), (1,12,2,11)(3,10,4,9)(5,14,6,13)(7,16,8,15)}

{(1,13,2,14)(3,15,4,16)(5,9,6,10)(7,11,8,12), (1,15,2,16)(3,13,4,14)(5,11,6,12)(7,9,8,10)}

{(1,14,2,13)(3,16,4,15)(5,10,6,9)(7,12,8,11), (1,16,2,15)(3,14,4,13)(5,12,6,11)(7,10,8,9)}