Properties

 Label 16T50 Order $$32$$ n $$16$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $SD_{16}:C_2$

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Group action invariants

 Degree $n$ : $16$ Transitive number $t$ : $50$ Group : $SD_{16}:C_2$ Parity: $1$ Primitive: No Nilpotency class: $3$ Generators: (1,8,2,7)(3,6,4,5)(9,12,10,11)(13,15,14,16), (9,10)(11,12)(13,14)(15,16), (1,13)(2,14)(3,11)(4,12)(5,15)(6,16)(7,10)(8,9) $|\Aut(F/K)|$: $8$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$
16:  $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $D_4$

Low degree siblings

16T32, 32T18

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $2$ $2$ $( 9,10)(11,12)(13,14)(15,16)$ $2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ $4, 4, 4, 4$ $2$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,15,10,16)(11,14,12,13)$ $4, 4, 4, 4$ $2$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,16,10,15)(11,13,12,14)$ $4, 4, 4, 4$ $4$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,13,10,14)(11,15,12,16)$ $4, 4, 4, 4$ $4$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,14,10,13)(11,16,12,15)$ $8, 8$ $4$ $8$ $( 1, 9, 3,16, 2,10, 4,15)( 5,11, 7,13, 6,12, 8,14)$ $8, 8$ $4$ $8$ $( 1, 9, 4,15, 2,10, 3,16)( 5,11, 8,14, 6,12, 7,13)$ $4, 4, 4, 4$ $4$ $4$ $( 1,11, 2,12)( 3,14, 4,13)( 5,10, 6, 9)( 7,16, 8,15)$ $2, 2, 2, 2, 2, 2, 2, 2$ $4$ $2$ $( 1,11)( 2,12)( 3,14)( 4,13)( 5,10)( 6, 9)( 7,16)( 8,15)$

Group invariants

 Order: $32=2^{5}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [32, 44]
 Character table:  2 5 4 5 4 4 3 3 3 3 3 3 1a 2a 2b 4a 4b 4c 4d 8a 8b 4e 2c 2P 1a 1a 1a 2b 2b 2b 2b 4b 4b 2b 1a 3P 1a 2a 2b 4a 4b 4c 4d 8a 8b 4e 2c 5P 1a 2a 2b 4a 4b 4c 4d 8a 8b 4e 2c 7P 1a 2a 2b 4a 4b 4c 4d 8a 8b 4e 2c X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 -1 1 -1 1 -1 1 X.3 1 -1 1 -1 1 -1 1 1 -1 1 -1 X.4 1 -1 1 -1 1 1 -1 -1 1 1 -1 X.5 1 -1 1 -1 1 1 -1 1 -1 -1 1 X.6 1 1 1 1 1 -1 -1 -1 -1 1 1 X.7 1 1 1 1 1 -1 -1 1 1 -1 -1 X.8 1 1 1 1 1 1 1 -1 -1 -1 -1 X.9 2 2 2 -2 -2 . . . . . . X.10 2 -2 2 2 -2 . . . . . . X.11 4 . -4 . . . . . . . .