Group action invariants
| Degree $n$: | $32$ | |
| Transitive number $t$: | $46$ | |
| Group: | $C_4.Q_8$ | |
| Parity: | $1$ | |
| Primitive: | no | |
| Nilpotency class: | $2$ | |
| $|\Aut(F/K)|$: | $32$ | |
| Generators: | (1,22,4,23)(2,21,3,24)(5,19,8,18)(6,20,7,17)(9,13,12,16)(10,14,11,15)(25,31,28,30)(26,32,27,29), (1,6,3,8)(2,5,4,7)(9,31,11,29)(10,32,12,30)(13,25,15,27)(14,26,16,28)(17,24,19,22)(18,23,20,21), (1,13,4,16)(2,14,3,15)(5,28,8,25)(6,27,7,26)(9,21,12,24)(10,22,11,23)(17,31,20,30)(18,32,19,29) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $8$: $C_2^3$, $Q_8$ x 2 $16$: $Q_8:C_2$ x 2, $Q_8\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 7
Degree 4: $C_2^2$ x 7
Degree 8: $C_2^3$, $Q_8$ x 2, $Q_8:C_2$ x 6
Degree 16: $Q_8\times C_2$, $Q_8 : C_2$ 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 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23) (22,24)(25,27)(26,28)(29,31)(30,32)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 5, 3, 7)( 2, 6, 4, 8)( 9,32,11,30)(10,31,12,29)(13,26,15,28)(14,25,16,27) (17,23,19,21)(18,24,20,22)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 9, 2,10)( 3,11, 4,12)( 5,31, 6,32)( 7,29, 8,30)(13,22,14,21)(15,24,16,23) (17,25,18,26)(19,27,20,28)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,11, 2,12)( 3, 9, 4,10)( 5,29, 6,30)( 7,31, 8,32)(13,24,14,23)(15,22,16,21) (17,27,18,28)(19,25,20,26)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,13, 4,16)( 2,14, 3,15)( 5,28, 8,25)( 6,27, 7,26)( 9,21,12,24)(10,22,11,23) (17,31,20,30)(18,32,19,29)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,17, 3,19)( 2,18, 4,20)( 5,22, 7,24)( 6,21, 8,23)( 9,25,11,27)(10,26,12,28) (13,30,15,32)(14,29,16,31)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,18, 3,20)( 2,17, 4,19)( 5,21, 7,23)( 6,22, 8,24)( 9,26,11,28)(10,25,12,27) (13,29,15,31)(14,30,16,32)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,21, 4,24)( 2,22, 3,23)( 5,20, 8,17)( 6,19, 7,18)( 9,14,12,15)(10,13,11,16) (25,32,28,29)(26,31,27,30)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,25, 4,28)( 2,26, 3,27)( 5,14, 8,15)( 6,13, 7,16)( 9,18,12,19)(10,17,11,20) (21,30,24,31)(22,29,23,32)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,26, 4,27)( 2,25, 3,28)( 5,13, 8,16)( 6,14, 7,15)( 9,17,12,20)(10,18,11,19) (21,29,24,32)(22,30,23,31)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,29, 3,31)( 2,30, 4,32)( 5, 9, 7,11)( 6,10, 8,12)(13,19,15,17)(14,20,16,18) (21,27,23,25)(22,28,24,26)$ |
Group invariants
| Order: | $32=2^{5}$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [32, 32] |
| Character table: |
2 5 5 5 5 3 4 4 3 4 4 3 4 4 3
1a 2a 2b 2c 4a 4b 4c 4d 4e 4f 4g 4h 4i 4j
2P 1a 1a 1a 1a 2b 2a 2a 2c 2b 2b 2c 2c 2c 2b
3P 1a 2a 2b 2c 4a 4b 4c 4d 4f 4e 4g 4i 4h 4j
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 1 -1 1 1 1 -1 -1 1 -1 -1 -1
X.6 1 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1
X.7 1 1 1 1 1 -1 -1 1 -1 -1 -1 1 1 -1
X.8 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1
X.9 2 2 -2 -2 . 2 -2 . . . . . . .
X.10 2 2 -2 -2 . -2 2 . . . . . . .
X.11 2 -2 -2 2 . . . . A -A . . . .
X.12 2 -2 -2 2 . . . . -A A . . . .
X.13 2 -2 2 -2 . . . . . . . A -A .
X.14 2 -2 2 -2 . . . . . . . -A A .
A = -2*E(4)
= -2*Sqrt(-1) = -2i
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