# Properties

 Label 16T20 Order $$32$$ n $$16$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $(C_2 \times Q_8):C_2$

# Related objects

## Group action invariants

 Degree $n$ : $16$ Transitive number $t$ : $20$ Group : $(C_2 \times Q_8):C_2$ Parity: $1$ Primitive: No Nilpotency class: $2$ Generators: (1,7,2,8)(3,6,4,5)(9,12,10,11)(13,15,14,16), (1,2)(3,4)(5,6)(7,8), (1,4,2,3)(5,7,6,8)(9,16,10,15)(11,13,12,14), (1,16)(2,15)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14) $|\Aut(F/K)|$: $8$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 15
4:  $C_2^2$ x 35
8:  $C_2^3$ x 15
16:  $C_2^4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

Degree 8: $C_2^3$

## Low degree siblings

16T20 x 4, 32T6

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [32, 50]
 Character table:  2 5 4 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 1a 2a 2b 4a 4b 4c 4d 4e 4f 2c 4g 4h 2d 2e 4i 4j 2f 2P 1a 1a 1a 2b 2b 2b 2b 2b 2b 1a 2b 2b 1a 1a 2b 2b 1a 3P 1a 2a 2b 4a 4b 4c 4d 4e 4f 2c 4g 4h 2d 2e 4i 4j 2f X.1 1 1 1 1 1 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 1 -1 1 -1 1 -1 X.3 1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 1 -1 1 -1 1 X.4 1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 X.5 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 X.6 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 X.7 1 -1 1 1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 1 -1 X.8 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 1 X.9 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 X.10 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 1 X.11 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 X.12 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 X.13 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 X.14 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 X.15 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 1 1 X.16 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 X.17 4 . -4 . . . . . . . . . . . . . .