Properties

Label 16T867
Order \(512\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes

Related objects

Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $867$
CHM label :  $t16n867$
Parity:  $1$
Primitive:  No
Generators:   ( 5,12, 7,10)( 6,11, 8, 9)(13,15)(14,16), ( 1,12,14, 6)( 2,11,13, 5)( 3, 9,15, 7)( 4,10,16, 8), ( 1,13, 3,15)( 2,14, 4,16)( 5,12, 7, 9)( 6,11, 8,10)
$|\textrm{Aut}(F/K)|$:  $2$
Low degree resolvents:  
2: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1
4: 4T1, 4T1, 4T1, 4T1, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2
8: 4T3, 4T3, 4T3, 4T3, 4T3, 4T3, 4T3, 4T3, 8T2, 8T2, 8T2, 8T2, 8T2, 8T2, 8T3
16: 8T9, 8T9, 8T9, 8T9, 8T10, 8T10, 8T10, 8T10, 8T11, 8T11, 16T2
32: 8T18, 8T21, 8T21, 8T21, 8T21, 16T19, 16T19, 16T21, 16T34, 16T34, 16T54
64: 8T31, 8T31, 16T76, 16T76, 32T?
128: 16T208, 16T237, 16T318
256: 32T?

Subfields

Degree 2: $C_2$

Degree 4: $D_4$

Degree 8: 8T29

Low degree siblings

16T824a, 16T824b, 16T824c, 16T824d, 16T867b, 16T867c, 16T867d, 16T915a, 16T915b, 16T926a, 16T926b
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1,14)( 2,13)( 3,15)( 4,16)( 5,11)( 6,12)( 7, 9)( 8,10)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 1, 2)( 3, 4)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$
$ 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 $ $32$ $4$ $( 1,12,14, 6)( 2,11,13, 5)( 3, 9,15, 7)( 4,10,16, 8)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1, 6,14,12)( 2, 5,13,11)( 3, 7,15, 9)( 4, 8,16,10)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 5, 8)( 6, 7)( 9,12)(10,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1,14)( 2,13)( 3,15)( 4,16)( 5,10)( 6, 9)( 7,12)( 8,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1,15)( 2,16)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,16)(14,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,14, 2,13)( 3,15, 4,16)( 5,11, 6,12)( 7, 9, 8,10)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 5, 8)( 6, 7)( 9,11)(10,12)(13,14)(15,16)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,14, 2,13)( 3,15, 4,16)( 5,10, 6, 9)( 7,12, 8,11)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $16$ $4$ $( 3, 4)( 7, 8)( 9,12,10,11)(13,16,14,15)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1,14, 3,16)( 2,13, 4,15)( 5,11, 7,10)( 6,12, 8, 9)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1,12,16, 6)( 2,11,15, 5)( 3,10,13, 8)( 4, 9,14, 7)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1, 6,14, 9)( 2, 5,13,10)( 3, 8,15,11)( 4, 7,16,12)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $16$ $4$ $( 3, 4)( 5, 8, 6, 7)(11,12)(13,16,14,15)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1,14, 3,16)( 2,13, 4,15)( 5,10, 8,12)( 6, 9, 7,11)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $16$ $4$ $( 5,12, 7,10)( 6,11, 8, 9)(13,15)(14,16)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1,14, 4,16)( 2,13, 3,15)( 5, 6)( 7, 8)( 9,12)(10,11)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 3)( 2, 4)( 5,10, 7,12)( 6, 9, 8,11)(13,14)(15,16)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $16$ $4$ $( 1,14, 4,16)( 2,13, 3,15)( 9,11)(10,12)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1,12, 3, 9)( 2,11, 4,10)( 5,14, 8,15)( 6,13, 7,16)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,14,10,13)(11,16,12,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,15)(10,16)(11,13)(12,14)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1,12, 3, 9)( 2,11, 4,10)( 5,13, 8,16)( 6,14, 7,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,14)(10,13)(11,16)(12,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 8, 2, 7)( 3, 5, 4, 6)( 9,15,10,16)(11,13,12,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $16$ $2$ $( 3, 4)( 5,10)( 6, 9)( 7,11)( 8,12)(15,16)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5, 8, 6, 7)( 9,12,10,11)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $16$ $4$ $( 1,15, 2,16)( 3,13, 4,14)( 5, 6)( 9,10)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 3, 2, 4)( 5,12, 6,11)( 7, 9, 8,10)(13,16,14,15)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1,12, 4, 9)( 2,11, 3,10)( 5,16, 7,13)( 6,15, 8,14)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 6, 3, 8)( 2, 5, 4, 7)( 9,14,12,16)(10,13,11,15)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1,12, 2,11)( 3,10, 4, 9)( 5,13)( 6,14)( 7,16)( 8,15)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,15,10,16)(11,14,12,13)$

Group invariants

Order:  $512=2^{9}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
Character table:  
      2  9  6  8  8  8  9  4  4  7  7  7  7  7  6  7  6  6  5  5  4  4  5  5

        1a 2a 2b 2c 2d 2e 4a 4b 2f 2g 2h 2i 2j 4c 2k 2l 4d 4e 4f 4g 4h 4i 4j
     2P 1a 1a 1a 1a 1a 1a 2a 2a 1a 1a 1a 1a 1a 2e 1a 1a 2e 2j 2k 2h 2g 2j 2k
     3P 1a 2a 2b 2c 2d 2e 4b 4a 2f 2g 2h 2i 2j 4c 2k 2l 4d 4i 4j 4g 4h 4e 4f

X.1      1  1  1  1  1  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 -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 -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  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  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 -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 -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  1  1  1  1  1  1
X.9      1 -1  1  1  1  1  A -A -1  1  1 -1 -1  1 -1  1 -1  A -A -1  1 -A  A
X.10     1 -1  1  1  1  1 -A  A -1  1  1 -1 -1  1 -1  1 -1 -A  A -1  1  A -A
X.11     1 -1  1  1  1  1  A -A -1  1  1 -1 -1  1 -1  1 -1  A -A -1  1 -A  A
X.12     1 -1  1  1  1  1 -A  A -1  1  1 -1 -1  1 -1  1 -1 -A  A -1  1  A -A
X.13     1 -1  1  1  1  1  A -A -1  1  1 -1 -1  1 -1  1 -1 -A  A  1 -1  A -A
X.14     1 -1  1  1  1  1 -A  A -1  1  1 -1 -1  1 -1  1 -1  A -A  1 -1 -A  A
X.15     1 -1  1  1  1  1  A -A -1  1  1 -1 -1  1 -1  1 -1 -A  A  1 -1  A -A
X.16     1 -1  1  1  1  1 -A  A -1  1  1 -1 -1  1 -1  1 -1  A -A  1 -1 -A  A
X.17     2 -2  2  2  2  2  .  . -2  2  2 -2  2 -2  2 -2  2  .  .  .  .  .  .
X.18     2 -2  2  2  2  2  .  . -2  2  2 -2  2 -2  2 -2  2  .  .  .  .  .  .
X.19     2  2  2  2  2  2  .  .  2  2  2  2 -2 -2 -2 -2 -2  .  .  .  .  .  .
X.20     2  2  2  2  2  2  .  .  2  2  2  2 -2 -2 -2 -2 -2  .  .  .  .  .  .
X.21     2 -2  2  2  2  2  .  .  2 -2 -2  2  2 -2  2  2 -2 -2  2  .  . -2  2
X.22     2 -2  2  2  2  2  .  .  2 -2 -2  2  2 -2  2  2 -2  2 -2  .  .  2 -2
X.23     2  2  2  2  2  2  .  . -2 -2 -2 -2  2  2  2 -2 -2  .  .  .  .  .  .
X.24     2  2  2  2  2  2  .  . -2 -2 -2 -2  2  2  2 -2 -2  .  .  .  .  .  .
X.25     2 -2  2  2  2  2  .  .  2 -2 -2  2 -2  2 -2 -2  2  .  .  .  .  .  .
X.26     2 -2  2  2  2  2  .  .  2 -2 -2  2 -2  2 -2 -2  2  .  .  .  .  .  .
X.27     2  2  2  2  2  2  .  . -2 -2 -2 -2 -2 -2 -2  2  2  B  B  .  . -B -B
X.28     2  2  2  2  2  2  .  . -2 -2 -2 -2 -2 -2 -2  2  2 -B -B  .  .  B  B
X.29     4  . -4 -4  4  4  .  .  . -4  4  .  .  .  .  .  .  .  .  .  .  .  .
X.30     4  . -4 -4  4  4  .  .  . -4  4  .  .  .  .  .  .  .  .  .  .  .  .
X.31     4  . -4 -4  4  4  .  .  .  4 -4  .  .  .  .  .  .  .  .  .  .  .  .
X.32     4  . -4 -4  4  4  .  .  .  4 -4  .  .  .  .  .  .  .  .  .  .  .  .
X.33     4  . -4  4 -4  4  .  .  .  .  .  . -4  .  4  .  .  .  .  .  .  .  .
X.34     4  . -4  4 -4  4  .  .  .  .  .  . -4  .  4  .  .  .  .  .  .  .  .
X.35     4  . -4  4 -4  4  .  .  .  .  .  .  4  . -4  .  .  .  .  .  .  .  .
X.36     4  . -4  4 -4  4  .  .  .  .  .  .  4  . -4  .  .  .  .  .  .  .  .
X.37     8  .  8 -8 -8  8  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
X.38     8  .  .  .  . -8  .  .  4  .  . -4  .  .  .  .  .  .  .  .  .  .  .
X.39     8  .  .  .  . -8  .  .  4  .  . -4  .  .  .  .  .  .  .  .  .  .  .
X.40     8  .  .  .  . -8  .  . -4  .  .  4  .  .  .  .  .  .  .  .  .  .  .
X.41     8  .  .  .  . -8  .  . -4  .  .  4  .  .  .  .  .  .  .  .  .  .  .

      2  5  5  5  5  5  6  6  5  6  6  5  5  5  5  5  5  5  5

        4k 4l 4m 4n 4o 4p 2m 4q 2n 4r 2o 4s 4t 4u 4v 4w 4x 4y
     2P 2f 2f 2f 2f 2d 2e 1a 2d 1a 2e 1a 2c 2c 2e 2k 2k 2j 2j
     3P 4n 4m 4l 4k 4o 4p 2m 4q 2n 4r 2o 4s 4t 4u 4w 4v 4y 4x

X.1      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 -1
X.3      1  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  1
X.5      1  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  1
X.7      1  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 -1
X.9      A -A  A -A -1  1  1  1 -1 -1 -1  1  1 -1 -A  A  A -A
X.10    -A  A -A  A -1  1  1  1 -1 -1 -1  1  1 -1  A -A -A  A
X.11    -A  A -A  A  1 -1 -1 -1  1  1  1 -1 -1  1  A -A -A  A
X.12     A -A  A -A  1 -1 -1 -1  1  1  1 -1 -1  1 -A  A  A -A
X.13     A -A  A -A -1  1  1  1 -1 -1  1 -1 -1  1  A -A -A  A
X.14    -A  A -A  A -1  1  1  1 -1 -1  1 -1 -1  1 -A  A  A -A
X.15    -A  A -A  A  1 -1 -1 -1  1  1 -1  1  1 -1 -A  A  A -A
X.16     A -A  A -A  1 -1 -1 -1  1  1 -1  1  1 -1  A -A -A  A
X.17     .  .  .  . -2  2  2 -2  2  2  .  .  .  .  .  .  .  .
X.18     .  .  .  .  2 -2 -2  2 -2 -2  .  .  .  .  .  .  .  .
X.19     .  .  .  . -2 -2 -2  2  2  2  .  .  .  .  .  .  .  .
X.20     .  .  .  .  2  2  2 -2 -2 -2  .  .  .  .  .  .  .  .
X.21     .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
X.22     .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
X.23     .  .  .  .  .  .  .  .  .  .  .  .  .  . -2 -2  2  2
X.24     .  .  .  .  .  .  .  .  .  .  .  .  .  .  2  2 -2 -2
X.25     .  .  .  .  .  .  .  .  .  .  .  .  .  .  B -B  B -B
X.26     .  .  .  .  .  .  .  .  .  .  .  .  .  . -B  B -B  B
X.27     .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
X.28     .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
X.29     .  .  .  .  . -4  4  .  .  .  .  .  .  .  .  .  .  .
X.30     .  .  .  .  .  4 -4  .  .  .  .  .  .  .  .  .  .  .
X.31     .  .  .  .  .  .  .  . -4  4  .  .  .  .  .  .  .  .
X.32     .  .  .  .  .  .  .  .  4 -4  .  .  .  .  .  .  .  .
X.33     .  .  .  .  .  .  .  .  .  . -2 -2  2  2  .  .  .  .
X.34     .  .  .  .  .  .  .  .  .  .  2  2 -2 -2  .  .  .  .
X.35     .  .  .  .  .  .  .  .  .  . -2  2 -2  2  .  .  .  .
X.36     .  .  .  .  .  .  .  .  .  .  2 -2  2 -2  .  .  .  .
X.37     .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
X.38    -2  2  2 -2  .  .  .  .  .  .  .  .  .  .  .  .  .  .
X.39     2 -2 -2  2  .  .  .  .  .  .  .  .  .  .  .  .  .  .
X.40     B  B -B -B  .  .  .  .  .  .  .  .  .  .  .  .  .  .
X.41    -B -B  B  B  .  .  .  .  .  .  .  .  .  .  .  .  .  .

A = -E(4)
  = -Sqrt(-1) = -i
B = -2*E(4)
  = -2*Sqrt(-1) = -2i