Group action invariants
| Degree $n$ : | $32$ | |
| Transitive number $t$ : | $47$ | |
| Group : | $C_8:C_4$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $3$ | |
| Generators: | (1,24,11,16)(2,23,12,15)(3,22,9,14)(4,21,10,13)(5,28,29,18)(6,27,30,17)(7,26,31,20)(8,25,32,19), (1,8,12,31)(2,7,11,32)(3,6,10,29)(4,5,9,30)(13,20,22,25)(14,19,21,26)(15,17,24,28)(16,18,23,27) | |
| $|\Aut(F/K)|$: | $32$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_4$ x 2, $C_2^2$ 8: $D_{4}$, $C_4\times C_2$, $Q_8$ 16: $QD_{16}$ x 2, $C_4:C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 2
Degree 8: $C_4\times C_2$, $D_4$, $Q_8$, $QD_{16}$ x 2
Low degree siblings
There are no siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)(17,20,18,19)(21,23,22,24) (25,27,26,28)(29,31,30,32)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 5,12,30)( 2, 6,11,29)( 3, 8,10,31)( 4, 7, 9,32)(13,18,22,27)(14,17,21,28) (15,20,24,25)(16,19,23,26)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 9, 2,10)( 3,12, 4,11)( 5,31, 6,32)( 7,30, 8,29)(13,23,14,24)(15,22,16,21) (17,26,18,25)(19,27,20,28)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,11)( 2,12)( 3, 9)( 4,10)( 5,29)( 6,30)( 7,31)( 8,32)(13,21)(14,22)(15,23) (16,24)(17,27)(18,28)(19,25)(20,26)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,12)( 2,11)( 3,10)( 4, 9)( 5,30)( 6,29)( 7,32)( 8,31)(13,22)(14,21)(15,24) (16,23)(17,28)(18,27)(19,26)(20,25)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,13,11,21)( 2,14,12,22)( 3,16, 9,24)( 4,15,10,23)( 5,19,29,25)( 6,20,30,26) ( 7,18,31,28)( 8,17,32,27)$ |
| $ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,17, 3,20, 2,18, 4,19)( 5,24, 7,21, 6,23, 8,22)( 9,26,12,28,10,25,11,27) (13,30,15,32,14,29,16,31)$ |
| $ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,18, 3,19, 2,17, 4,20)( 5,23, 7,22, 6,24, 8,21)( 9,25,12,27,10,26,11,28) (13,29,15,31,14,30,16,32)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,21,11,13)( 2,22,12,14)( 3,24, 9,16)( 4,23,10,15)( 5,25,29,19)( 6,26,30,20) ( 7,28,31,18)( 8,27,32,17)$ |
| $ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,25, 4,28, 2,26, 3,27)( 5,14, 8,15, 6,13, 7,16)( 9,17,11,19,10,18,12,20) (21,31,24,29,22,32,23,30)$ |
| $ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,26, 4,27, 2,25, 3,28)( 5,13, 8,16, 6,14, 7,15)( 9,18,11,20,10,17,12,19) (21,32,24,30,22,31,23,29)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,29,12, 6)( 2,30,11, 5)( 3,32,10, 7)( 4,31, 9, 8)(13,28,22,17)(14,27,21,18) (15,26,24,19)(16,25,23,20)$ |
Group invariants
| Order: | $32=2^{5}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [32, 13] |
| Character table: |
2 5 5 4 3 4 5 5 3 4 4 3 4 4 3
1a 2a 4a 4b 4c 2b 2c 4d 8a 8b 4e 8c 8d 4f
2P 1a 1a 2a 2c 2a 1a 1a 2b 4a 4a 2b 4a 4a 2c
3P 1a 2a 4a 4f 4c 2b 2c 4e 8a 8b 4d 8c 8d 4b
5P 1a 2a 4a 4b 4c 2b 2c 4d 8b 8a 4e 8d 8c 4f
7P 1a 2a 4a 4f 4c 2b 2c 4e 8b 8a 4d 8d 8c 4b
X.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
X.3 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
X.5 1 1 1 A -1 -1 -1 A -1 -1 -A 1 1 -A
X.6 1 1 1 -A -1 -1 -1 -A -1 -1 A 1 1 A
X.7 1 1 1 A -1 -1 -1 -A 1 1 A -1 -1 -A
X.8 1 1 1 -A -1 -1 -1 A 1 1 -A -1 -1 A
X.9 2 2 -2 . -2 2 2 . . . . . . .
X.10 2 2 -2 . 2 -2 -2 . . . . . . .
X.11 2 -2 . . . -2 2 . B -B . B -B .
X.12 2 -2 . . . -2 2 . -B B . -B B .
X.13 2 -2 . . . 2 -2 . B -B . -B B .
X.14 2 -2 . . . 2 -2 . -B B . B -B .
A = -E(4)
= -Sqrt(-1) = -i
B = -E(8)-E(8)^3
= -Sqrt(-2) = -i2
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