Group action invariants
| Degree $n$ : | $32$ | |
| Transitive number $t$ : | $44$ | |
| Group : | $C_4:C_8$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $2$ | |
| Generators: | (1,5,9,13,17,23,25,29)(2,6,10,14,18,24,26,30)(3,8,11,16,19,22,27,32)(4,7,12,15,20,21,28,31), (1,22,9,32,17,8,25,16)(2,21,10,31,18,7,26,15)(3,24,11,30,19,6,27,14)(4,23,12,29,20,5,28,13) | |
| $|\Aut(F/K)|$: | $32$ |
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: $D_{4}$, $C_8$ x 2, $C_4\times C_2$, $Q_8$ 16: $C_8:C_2$, $C_8\times C_2$, $C_4:C_4$ 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: $C_8$ x 2, $C_4\times C_2$, $D_4$, $Q_8$, $C_8:C_2$
Degree 16: $C_8\times C_2$, $C_8: C_2$, $C_4:C_4$
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 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)(17,19,18,20)(21,24,22,23) (25,27,26,28)(29,31,30,32)$ |
| $ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1, 5, 9,13,17,23,25,29)( 2, 6,10,14,18,24,26,30)( 3, 8,11,16,19,22,27,32) ( 4, 7,12,15,20,21,28,31)$ |
| $ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1, 7, 9,15,17,21,25,31)( 2, 8,10,16,18,22,26,32)( 3, 5,11,13,19,23,27,29) ( 4, 6,12,14,20,24,28,30)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 9,17,25)( 2,10,18,26)( 3,11,19,27)( 4,12,20,28)( 5,13,23,29)( 6,14,24,30) ( 7,15,21,31)( 8,16,22,32)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,10,17,26)( 2, 9,18,25)( 3,12,19,28)( 4,11,20,27)( 5,14,23,30)( 6,13,24,29) ( 7,16,21,32)( 8,15,22,31)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,11,18,28)( 2,12,17,27)( 3,10,20,25)( 4, 9,19,26)( 5,15,24,32)( 6,16,23,31) ( 7,14,22,29)( 8,13,21,30)$ |
| $ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,13,25, 5,17,29, 9,23)( 2,14,26, 6,18,30,10,24)( 3,16,27, 8,19,32,11,22) ( 4,15,28, 7,20,31,12,21)$ |
| $ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,15,25, 7,17,31, 9,21)( 2,16,26, 8,18,32,10,22)( 3,13,27, 5,19,29,11,23) ( 4,14,28, 6,20,30,12,24)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,17)( 2,18)( 3,19)( 4,20)( 5,23)( 6,24)( 7,21)( 8,22)( 9,25)(10,26)(11,27) (12,28)(13,29)(14,30)(15,31)(16,32)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,18)( 2,17)( 3,20)( 4,19)( 5,24)( 6,23)( 7,22)( 8,21)( 9,26)(10,25)(11,28) (12,27)(13,30)(14,29)(15,32)(16,31)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,19, 2,20)( 3,18, 4,17)( 5,21, 6,22)( 7,24, 8,23)( 9,27,10,28)(11,26,12,25) (13,31,14,32)(15,30,16,29)$ |
| $ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,21, 9,31,17, 7,25,15)( 2,22,10,32,18, 8,26,16)( 3,23,11,29,19, 5,27,13) ( 4,24,12,30,20, 6,28,14)$ |
| $ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,23, 9,29,17, 5,25,13)( 2,24,10,30,18, 6,26,14)( 3,22,11,32,19, 8,27,16) ( 4,21,12,31,20, 7,28,15)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,25,17, 9)( 2,26,18,10)( 3,27,19,11)( 4,28,20,12)( 5,29,23,13)( 6,30,24,14) ( 7,31,21,15)( 8,32,22,16)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,26,17,10)( 2,25,18, 9)( 3,28,19,12)( 4,27,20,11)( 5,30,23,14)( 6,29,24,13) ( 7,32,21,16)( 8,31,22,15)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,27,18,12)( 2,28,17,11)( 3,26,20, 9)( 4,25,19,10)( 5,31,24,16)( 6,32,23,15) ( 7,30,22,13)( 8,29,21,14)$ |
| $ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,29,25,23,17,13, 9, 5)( 2,30,26,24,18,14,10, 6)( 3,32,27,22,19,16,11, 8) ( 4,31,28,21,20,15,12, 7)$ |
| $ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,31,25,21,17,15, 9, 7)( 2,32,26,22,18,16,10, 8)( 3,29,27,23,19,13,11, 5) ( 4,30,28,24,20,14,12, 6)$ |
Group invariants
| Order: | $32=2^{5}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [32, 12] |
| Character table: |
2 5 5 4 4 4 5 5 4 4 4 5 5 4 4 4 5 5 4 4 4
1a 2a 4a 8a 8b 4b 4c 4d 8c 8d 2b 2c 4e 8e 8f 4f 4g 4h 8g 8h
2P 1a 1a 2a 4b 4b 2b 2b 2c 4f 4f 1a 1a 2a 4b 4b 2b 2b 2c 4f 4f
3P 1a 2a 4a 8c 8d 4f 4g 4h 8a 8b 2b 2c 4e 8h 8g 4b 4c 4d 8f 8e
5P 1a 2a 4a 8f 8e 4b 4c 4d 8g 8h 2b 2c 4e 8b 8a 4f 4g 4h 8c 8d
7P 1a 2a 4a 8g 8h 4f 4g 4h 8f 8e 2b 2c 4e 8d 8c 4b 4c 4d 8a 8b
X.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
X.3 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
X.5 1 1 -1 A -A -1 -1 1 -A A 1 1 -1 -A A -1 -1 1 -A A
X.6 1 1 -1 -A A -1 -1 1 A -A 1 1 -1 A -A -1 -1 1 A -A
X.7 1 1 -1 B -B -A -A A -/B /B -1 -1 1 B -B A A -A /B -/B
X.8 1 1 -1 -/B /B A A -A B -B -1 -1 1 -/B /B -A -A A -B B
X.9 1 1 -1 /B -/B A A -A -B B -1 -1 1 /B -/B -A -A A B -B
X.10 1 1 -1 -B B -A -A A /B -/B -1 -1 1 -B B A A -A -/B /B
X.11 1 1 1 A A -1 -1 -1 -A -A 1 1 1 A A -1 -1 -1 -A -A
X.12 1 1 1 -A -A -1 -1 -1 A A 1 1 1 -A -A -1 -1 -1 A A
X.13 1 1 1 B B -A -A -A -/B -/B -1 -1 -1 -B -B A A A /B /B
X.14 1 1 1 -/B -/B A A A B B -1 -1 -1 /B /B -A -A -A -B -B
X.15 1 1 1 /B /B A A A -B -B -1 -1 -1 -/B -/B -A -A -A B B
X.16 1 1 1 -B -B -A -A -A /B /B -1 -1 -1 B B A A A -/B -/B
X.17 2 -2 . . . -2 2 . . . 2 -2 . . . -2 2 . . .
X.18 2 -2 . . . 2 -2 . . . 2 -2 . . . 2 -2 . . .
X.19 2 -2 . . . C -C . . . -2 2 . . . -C C . . .
X.20 2 -2 . . . -C C . . . -2 2 . . . C -C . . .
A = -E(4)
= -Sqrt(-1) = -i
B = -E(8)
C = -2*E(4)
= -2*Sqrt(-1) = -2i
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