Group action invariants
| Degree $n$ : | $32$ | |
| Transitive number $t$ : | $48$ | |
| Group : | $C_2.D_8$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $3$ | |
| Generators: | (1,30,11,6)(2,29,12,5)(3,32,9,8)(4,31,10,7)(13,28,21,18)(14,27,22,17)(15,25,23,19)(16,26,24,20), (1,18,3,20,2,17,4,19)(5,24,8,21,6,23,7,22)(9,26,12,27,10,25,11,28)(13,30,15,31,14,29,16,32) | |
| $|\Aut(F/K)|$: | $32$ |
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: $D_{8}$, $C_4:C_4$, $Q_{16}$ 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: $C_4\times C_2$, $D_4$, $Q_8$, $D_{8}$ 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 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)(17,19,18,20)(21,23,22,24) (25,28,26,27)(29,32,30,31)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 5,11,29)( 2, 6,12,30)( 3, 7, 9,31)( 4, 8,10,32)(13,17,21,27)(14,18,22,28) (15,20,23,26)(16,19,24,25)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 9, 2,10)( 3,12, 4,11)( 5,32, 6,31)( 7,29, 8,30)(13,23,14,24)(15,22,16,21) (17,25,18,26)(19,28,20,27)$ |
| $ 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 $ | $4$ | $4$ | $( 1,13,11,21)( 2,14,12,22)( 3,16, 9,24)( 4,15,10,23)( 5,20,29,26)( 6,19,30,25) ( 7,17,31,27)( 8,18,32,28)$ |
| $ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,17, 3,19, 2,18, 4,20)( 5,23, 8,22, 6,24, 7,21)( 9,25,12,28,10,26,11,27) (13,29,15,32,14,30,16,31)$ |
| $ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,18, 3,20, 2,17, 4,19)( 5,24, 8,21, 6,23, 7,22)( 9,26,12,27,10,25,11,28) (13,30,15,31,14,29,16,32)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,21,11,13)( 2,22,12,14)( 3,24, 9,16)( 4,23,10,15)( 5,26,29,20)( 6,25,30,19) ( 7,27,31,17)( 8,28,32,18)$ |
| $ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,25, 4,27, 2,26, 3,28)( 5,14, 7,15, 6,13, 8,16)( 9,18,11,19,10,17,12,20) (21,32,24,29,22,31,23,30)$ |
| $ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,26, 4,28, 2,25, 3,27)( 5,13, 7,16, 6,14, 8,15)( 9,17,11,20,10,18,12,19) (21,31,24,30,22,32,23,29)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,29,11, 5)( 2,30,12, 6)( 3,31, 9, 7)( 4,32,10, 8)(13,27,21,17)(14,28,22,18) (15,26,23,20)(16,25,24,19)$ |
Group invariants
| Order: | $32=2^{5}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [32, 14] |
| 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 2b 2a 1a 1a 2b 4a 4a 2b 4a 4a 2b
3P 1a 2a 4a 4f 4c 2b 2c 4e 8b 8a 4d 8d 8c 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 8a 8b 4d 8c 8d 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) = -r2
|