Properties

Label 16T1194
Order \(1024\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes

Related objects

Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $1194$
CHM label :  $t16n1194$
Parity:  $1$
Primitive:  No
Generators:   ( 1,12, 5,16, 9, 3,14, 7)( 2,11, 6,15,10, 4,13, 8), ( 1,14,10, 5, 2,13, 9, 6)( 3,15,11, 7, 4,16,12, 8)
$|\Aut(F/K)|$:  $2$
Low degree resolvents:  
2: 2T1, 2T1, 2T1
4: 4T1, 4T1, 4T2
8: 4T3, 4T3, 8T1, 8T1, 8T2
16: 8T7, 8T10, 16T5
32: 8T16, 8T21, 16T24
64: 8T27, 8T27, 16T85
128: 16T258
256: 16T651
512: 16T817

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $C_8$

Low degree siblings

16T1194b, 16T1194c, 16T1194d, 16T1194e, 16T1194f, 16T1194g, 16T1194h, 16T1216a, 16T1216b, 16T1216c, 16T1216d, 16T1216e, 16T1216f, 16T1216g, 16T1216h
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 $ $8$ $2$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 3, 4)( 5, 6)(11,12)(13,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 3, 4)( 7, 8)(11,12)(15,16)$
$ 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 $ $32$ $4$ $( 1, 5, 9,14)( 2, 6,10,13)( 3, 7,12,16)( 4, 8,11,15)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1,14, 9, 5)( 2,13,10, 6)( 3,16,12, 7)( 4,15,11, 8)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 7, 8)( 9,10)(11,12)(13,14)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,14, 6,13)( 7,15, 8,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 5, 6)( 9,10)(11,12)(15,16)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 7, 8)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14)( 6,13)( 7,15)( 8,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 3, 4)( 5, 6)( 7, 8)(11,12)(13,14)(15,16)$
$ 8, 8 $ $64$ $8$ $( 1,12, 5,16, 9, 3,14, 7)( 2,11, 6,15,10, 4,13, 8)$
$ 8, 8 $ $64$ $8$ $( 1, 3, 5, 7, 9,12,14,16)( 2, 4, 6, 8,10,11,13,15)$
$ 8, 8 $ $64$ $8$ $( 1,16,14,12, 9, 7, 5, 3)( 2,15,13,11,10, 8, 6, 4)$
$ 8, 8 $ $64$ $8$ $( 1, 7,14, 3, 9,16, 5,12)( 2, 8,13, 4,10,15, 6,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $(11,12)(15,16)$
$ 4, 4, 2, 2, 2, 2 $ $32$ $4$ $( 1, 9)( 2,10)( 3,12, 4,11)( 5,14)( 6,13)( 7,16, 8,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 5, 6)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(13,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 7, 8)( 9,10)(11,12)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1, 5, 9,14)( 2, 6,10,13)( 3, 7,12,15)( 4, 8,11,16)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1,14, 9, 5)( 2,13,10, 6)( 3,16,11, 7)( 4,15,12, 8)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $(13,14)(15,16)$
$ 4, 4, 2, 2, 2, 2 $ $32$ $4$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14, 6,13)( 7,16, 8,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 5, 6)(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 7, 8)( 9,10)(13,14)$
$ 8, 8 $ $64$ $8$ $( 1, 5, 9,14, 2, 6,10,13)( 3, 7,12,16, 4, 8,11,15)$
$ 8, 8 $ $64$ $8$ $( 1,14,10, 6, 2,13, 9, 5)( 3,16,11, 8, 4,15,12, 7)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 7, 8)( 9,10)(11,12)(15,16)$
$ 4, 4, 2, 2, 2, 2 $ $32$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,14)( 6,13)( 7,15)( 8,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 5, 6)( 7, 8)( 9,10)(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 5, 6)( 9,10)(11,12)(13,14)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 9,10)$
$ 8, 8 $ $64$ $8$ $( 1,12, 5,16,10, 4,13, 7)( 2,11, 6,15, 9, 3,14, 8)$
$ 8, 8 $ $64$ $8$ $( 1, 3, 5, 7, 9,12,14,15)( 2, 4, 6, 8,10,11,13,16)$
$ 8, 8 $ $64$ $8$ $( 1,16,13,12, 9, 7, 5, 3)( 2,15,14,11,10, 8, 6, 4)$
$ 8, 8 $ $64$ $8$ $( 1, 7,14, 4,10,15, 5,12)( 2, 8,13, 3, 9,16, 6,11)$

Group invariants

Order:  $1024=2^{10}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table:  
      2 10  7  8  9 10  5  5  7  6  7  8  7  8   4   4   4   4  7  5  7  7  7

        1a 2a 2b 2c 2d 4a 4b 2e 4c 2f 2g 2h 2i  8a  8b  8c  8d 2j 4d 2k 2l 2m
     2P 1a 1a 1a 1a 1a 2a 2a 1a 2d 1a 1a 1a 1a  4a  4a  4b  4b 1a 2c 1a 1a 1a
     3P 1a 2a 2b 2c 2d 4b 4a 2e 4c 2f 2g 2h 2i  8c  8d  8a  8b 2j 4d 2k 2l 2m
     5P 1a 2a 2b 2c 2d 4a 4b 2e 4c 2f 2g 2h 2i  8b  8a  8d  8c 2j 4d 2k 2l 2m
     7P 1a 2a 2b 2c 2d 4b 4a 2e 4c 2f 2g 2h 2i  8d  8c  8b  8a 2j 4d 2k 2l 2m

X.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
X.3      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
X.5      1 -1  1  1  1  A -A  1 -1  1  1 -1  1   C  -C -/C  /C  1 -1  1  1  1
X.6      1 -1  1  1  1  A -A  1 -1  1  1 -1  1  -C   C  /C -/C  1 -1  1  1  1
X.7      1 -1  1  1  1 -A  A  1 -1  1  1 -1  1 -/C  /C   C  -C  1 -1  1  1  1
X.8      1 -1  1  1  1 -A  A  1 -1  1  1 -1  1  /C -/C  -C   C  1 -1  1  1  1
X.9      1 -1  1  1  1  A -A  1 -1  1  1 -1  1   C  -C -/C  /C  1 -1  1  1  1
X.10     1 -1  1  1  1  A -A  1 -1  1  1 -1  1  -C   C  /C -/C  1 -1  1  1  1
X.11     1 -1  1  1  1 -A  A  1 -1  1  1 -1  1 -/C  /C   C  -C  1 -1  1  1  1
X.12     1 -1  1  1  1 -A  A  1 -1  1  1 -1  1  /C -/C  -C   C  1 -1  1  1  1
X.13     1  1  1  1  1 -1 -1  1  1  1  1  1  1   A   A  -A  -A  1  1  1  1  1
X.14     1  1  1  1  1 -1 -1  1  1  1  1  1  1  -A  -A   A   A  1  1  1  1  1
X.15     1  1  1  1  1 -1 -1  1  1  1  1  1  1   A   A  -A  -A  1  1  1  1  1
X.16     1  1  1  1  1 -1 -1  1  1  1  1  1  1  -A  -A   A   A  1  1  1  1  1
X.17     2  2  2  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 -2 -2 -2
X.19     2 -2  2  2  2  B -B  2 -2  2  2 -2  2   .   .   .   . -2  2 -2 -2 -2
X.20     2 -2  2  2  2 -B  B  2 -2  2  2 -2  2   .   .   .   . -2  2 -2 -2 -2
X.21     4 -4  4  4  4  .  .  .  .  . -4  4 -4   .   .   .   .  .  .  .  .  .
X.22     4 -4  4  4  4  .  .  .  .  . -4  4 -4   .   .   .   .  .  .  .  .  .
X.23     4  4  4  4  4  .  .  .  .  . -4 -4 -4   .   .   .   .  .  .  .  .  .
X.24     4  4  4  4  4  .  .  .  .  . -4 -4 -4   .   .   .   .  .  .  .  .  .
X.25     4 -4  4  4  4  .  . -4  4 -4  4 -4  4   .   .   .   .  .  .  .  .  .
X.26     4  4  4  4  4  .  . -4 -4 -4  4  4  4   .   .   .   .  .  .  .  .  .
X.27     8  . -8  8  8  .  .  .  .  .  .  .  .   .   .   .   . -4  .  4 -4  4
X.28     8  . -8  8  8  .  .  .  .  .  .  .  .   .   .   .   .  4  . -4  4 -4
X.29     8  .  .  . -8  .  .  .  .  .  4  . -4   .   .   .   . -4  .  .  4  .
X.30     8  .  .  . -8  .  .  .  .  .  4  . -4   .   .   .   . -4  .  .  4  .
X.31     8  .  .  . -8  .  .  .  .  . -4  .  4   .   .   .   .  .  . -4  .  4
X.32     8  .  .  . -8  .  .  .  .  . -4  .  4   .   .   .   .  .  . -4  .  4
X.33     8  .  . -8  8  .  .  4  . -4  .  .  .   .   .   .   .  .  .  .  .  .
X.34     8  .  . -8  8  .  .  4  . -4  .  .  .   .   .   .   .  .  .  .  .  .
X.35     8  .  . -8  8  .  . -4  .  4  .  .  .   .   .   .   .  .  .  .  .  .
X.36     8  .  . -8  8  .  . -4  .  4  .  .  .   .   .   .   .  .  .  .  .  .
X.37     8  .  .  . -8  .  .  .  .  . -4  .  4   .   .   .   .  .  .  4  . -4
X.38     8  .  .  . -8  .  .  .  .  . -4  .  4   .   .   .   .  .  .  4  . -4
X.39     8  .  .  . -8  .  .  .  .  .  4  . -4   .   .   .   .  4  .  . -4  .
X.40     8  .  .  . -8  .  .  .  .  .  4  . -4   .   .   .   .  4  .  . -4  .

      2  5  5  7  5  7  7  7  4  4  7  5  7  7  7   4   4   4   4

        4e 4f 2n 4g 2o 2p 2q 8e 8f 2r 4h 2s 2t 2u  8g  8h  8i  8j
     2P 2h 2h 1a 2b 1a 1a 1a 4c 4c 1a 2b 1a 1a 1a  4e  4e  4f  4f
     3P 4f 4e 2n 4g 2o 2p 2q 8f 8e 2r 4h 2s 2t 2u  8i  8j  8g  8h
     5P 4e 4f 2n 4g 2o 2p 2q 8e 8f 2r 4h 2s 2t 2u  8h  8g  8j  8i
     7P 4f 4e 2n 4g 2o 2p 2q 8f 8e 2r 4h 2s 2t 2u  8j  8i  8h  8g

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      A -A -1  1 -1 -1 -1 -A  A -1  1 -1 -1 -1  -C   C  /C -/C
X.6      A -A -1  1 -1 -1 -1 -A  A -1  1 -1 -1 -1   C  -C -/C  /C
X.7     -A  A -1  1 -1 -1 -1  A -A -1  1 -1 -1 -1  /C -/C  -C   C
X.8     -A  A -1  1 -1 -1 -1  A -A -1  1 -1 -1 -1 -/C  /C   C  -C
X.9      A -A  1 -1  1  1  1  A -A  1 -1  1  1  1   C  -C -/C  /C
X.10     A -A  1 -1  1  1  1  A -A  1 -1  1  1  1  -C   C  /C -/C
X.11    -A  A  1 -1  1  1  1 -A  A  1 -1  1  1  1 -/C  /C   C  -C
X.12    -A  A  1 -1  1  1  1 -A  A  1 -1  1  1  1  /C -/C  -C   C
X.13    -1 -1 -1 -1 -1 -1 -1  1  1 -1 -1 -1 -1 -1  -A  -A   A   A
X.14    -1 -1 -1 -1 -1 -1 -1  1  1 -1 -1 -1 -1 -1   A   A  -A  -A
X.15    -1 -1  1  1  1  1  1 -1 -1  1  1  1  1  1   A   A  -A  -A
X.16    -1 -1  1  1  1  1  1 -1 -1  1  1  1  1  1  -A  -A   A   A
X.17     2  2  .  .  .  .  .  .  .  .  .  .  .  .   .   .   .   .
X.18    -2 -2  .  .  .  .  .  .  .  .  .  .  .  .   .   .   .   .
X.19    -B  B  .  .  .  .  .  .  .  .  .  .  .  .   .   .   .   .
X.20     B -B  .  .  .  .  .  .  .  .  .  .  .  .   .   .   .   .
X.21     .  . -2  2 -2 -2 -2  .  .  2 -2  2  2  2   .   .   .   .
X.22     .  .  2 -2  2  2  2  .  . -2  2 -2 -2 -2   .   .   .   .
X.23     .  . -2 -2 -2 -2 -2  .  .  2  2  2  2  2   .   .   .   .
X.24     .  .  2  2  2  2  2  .  . -2 -2 -2 -2 -2   .   .   .   .
X.25     .  .  .  .  .  .  .  .  .  .  .  .  .  .   .   .   .   .
X.26     .  .  .  .  .  .  .  .  .  .  .  .  .  .   .   .   .   .
X.27     .  .  .  .  .  .  .  .  .  .  .  .  .  .   .   .   .   .
X.28     .  .  .  .  .  .  .  .  .  .  .  .  .  .   .   .   .   .
X.29     .  .  .  . -4  .  4  .  . -4  .  .  4  .   .   .   .   .
X.30     .  .  .  .  4  . -4  .  .  4  .  . -4  .   .   .   .   .
X.31     .  .  .  . -4  .  4  .  .  4  .  . -4  .   .   .   .   .
X.32     .  .  .  .  4  . -4  .  . -4  .  .  4  .   .   .   .   .
X.33     .  .  4  . -4  4 -4  .  .  .  .  .  .  .   .   .   .   .
X.34     .  . -4  .  4 -4  4  .  .  .  .  .  .  .   .   .   .   .
X.35     .  .  .  .  .  .  .  .  . -4  .  4 -4  4   .   .   .   .
X.36     .  .  .  .  .  .  .  .  .  4  . -4  4 -4   .   .   .   .
X.37     .  . -4  .  .  4  .  .  .  .  . -4  .  4   .   .   .   .
X.38     .  .  4  .  . -4  .  .  .  .  .  4  . -4   .   .   .   .
X.39     .  . -4  .  .  4  .  .  .  .  .  4  . -4   .   .   .   .
X.40     .  .  4  .  . -4  .  .  .  .  . -4  .  4   .   .   .   .

A = -E(4)
  = -Sqrt(-1) = -i
B = -2*E(4)
  = -2*Sqrt(-1) = -2i
C = -E(8)^3