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 Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2$ $8$ $2$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $4$ $2$ $( 3, 4)( 5, 6)(11,12)(13,14)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $2$ $2$ $( 3, 4)( 7, 8)(11,12)(15,16)$ $2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ $4, 4, 4, 4$ $32$ $4$ $( 1, 5, 9,14)( 2, 6,10,13)( 3, 7,12,16)( 4, 8,11,15)$ $4, 4, 4, 4$ $32$ $4$ $( 1,14, 9, 5)( 2,13,10, 6)( 3,16,12, 7)( 4,15,11, 8)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $8$ $2$ $( 7, 8)( 9,10)(11,12)(13,14)$ $4, 4, 4, 4$ $16$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,14, 6,13)( 7,15, 8,16)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $8$ $2$ $( 5, 6)( 9,10)(11,12)(15,16)$ $2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $4$ $2$ $( 7, 8)(15,16)$ $2, 2, 2, 2, 2, 2, 2, 2$ $8$ $2$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14)( 6,13)( 7,15)( 8,16)$ $2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $4$ $2$ $( 3, 4)( 5, 6)( 7, 8)(11,12)(13,14)(15,16)$ $8, 8$ $64$ $8$ $( 1,12, 5,16, 9, 3,14, 7)( 2,11, 6,15,10, 4,13, 8)$ $8, 8$ $64$ $8$ $( 1, 3, 5, 7, 9,12,14,16)( 2, 4, 6, 8,10,11,13,15)$ $8, 8$ $64$ $8$ $( 1,16,14,12, 9, 7, 5, 3)( 2,15,13,11,10, 8, 6, 4)$ $8, 8$ $64$ $8$ $( 1, 7,14, 3, 9,16, 5,12)( 2, 8,13, 4,10,15, 6,11)$ $2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $8$ $2$ $(11,12)(15,16)$ $4, 4, 2, 2, 2, 2$ $32$ $4$ $( 1, 9)( 2,10)( 3,12, 4,11)( 5,14)( 6,13)( 7,16, 8,15)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $8$ $2$ $( 3, 4)( 5, 6)(13,14)(15,16)$ $2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $8$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(13,14)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $8$ $2$ $( 1, 2)( 7, 8)( 9,10)(11,12)$ $4, 4, 4, 4$ $32$ $4$ $( 1, 5, 9,14)( 2, 6,10,13)( 3, 7,12,15)( 4, 8,11,16)$ $4, 4, 4, 4$ $32$ $4$ $( 1,14, 9, 5)( 2,13,10, 6)( 3,16,11, 7)( 4,15,12, 8)$ $2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $8$ $2$ $(13,14)(15,16)$ $4, 4, 2, 2, 2, 2$ $32$ $4$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14, 6,13)( 7,16, 8,15)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $8$ $2$ $( 3, 4)( 5, 6)(11,12)(15,16)$ $2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $8$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $8$ $2$ $( 1, 2)( 7, 8)( 9,10)(13,14)$ $8, 8$ $64$ $8$ $( 1, 5, 9,14, 2, 6,10,13)( 3, 7,12,16, 4, 8,11,15)$ $8, 8$ $64$ $8$ $( 1,14,10, 6, 2,13, 9, 5)( 3,16,11, 8, 4,15,12, 7)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $8$ $2$ $( 7, 8)( 9,10)(11,12)(15,16)$ $4, 4, 2, 2, 2, 2$ $32$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,14)( 6,13)( 7,15)( 8,16)$ $2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $8$ $2$ $( 3, 4)( 5, 6)( 7, 8)( 9,10)(13,14)(15,16)$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $8$ $2$ $( 5, 6)( 9,10)(11,12)(13,14)$ $2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $8$ $2$ $( 3, 4)( 9,10)$ $8, 8$ $64$ $8$ $( 1,12, 5,16,10, 4,13, 7)( 2,11, 6,15, 9, 3,14, 8)$ $8, 8$ $64$ $8$ $( 1, 3, 5, 7, 9,12,14,15)( 2, 4, 6, 8,10,11,13,16)$ $8, 8$ $64$ $8$ $( 1,16,13,12, 9, 7, 5, 3)( 2,15,14,11,10, 8, 6, 4)$ $8, 8$ $64$ $8$ $( 1, 7,14, 4,10,15, 5,12)( 2, 8,13, 3, 9,16, 6,11)$

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 ```