# Properties

 Label 32T50 Order $$32$$ n $$32$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $Q_8:C_4$

## Group action invariants

 Degree $n$ : $32$ Transitive number $t$ : $50$ Group : $Q_8:C_4$ Parity: $1$ Primitive: No Nilpotency class: $3$ Generators: (1,10,2,9)(3,11,4,12)(5,14,6,13)(7,16,8,15)(17,25,18,26)(19,27,20,28)(21,29,22,30)(23,31,24,32), (1,23,11,15,2,24,12,16)(3,22,9,13,4,21,10,14)(5,26,30,20,6,25,29,19)(7,28,31,17,8,27,32,18) $|\Aut(F/K)|$: $32$

## 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:  $D_{4}$ x 2, $C_4\times C_2$
16:  $QD_{16}$, $C_2^2:C_4$, $Q_{16}$

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 4

Degree 8: $C_4\times C_2$, $D_4$ x 2, $QD_{16}$, $C_2^2:C_4$ x 2

Degree 16: $C_2^2 : C_4$, $QD_{16}$, $Q_{16}$

## 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)$ $4, 4, 4, 4, 4, 4, 4, 4$ $4$ $4$ $( 1, 3, 2, 4)( 5,22, 6,21)( 7,23, 8,24)( 9,11,10,12)(13,30,14,29)(15,31,16,32) (17,19,18,20)(25,28,26,27)$ $8, 8, 8, 8$ $2$ $8$ $( 1, 5,12,29, 2, 6,11,30)( 3, 8,10,31, 4, 7, 9,32)(13,18,22,27,14,17,21,28) (15,20,23,26,16,19,24,25)$ $8, 8, 8, 8$ $2$ $8$ $( 1, 6,12,30, 2, 5,11,29)( 3, 7,10,32, 4, 8, 9,31)(13,17,22,28,14,18,21,27) (15,19,23,25,16,20,24,26)$ $4, 4, 4, 4, 4, 4, 4, 4$ $4$ $4$ $( 1, 7,25,14)( 2, 8,26,13)( 3, 5,28,15)( 4, 6,27,16)( 9,29,17,23)(10,30,18,24) (11,32,19,21)(12,31,20,22)$ $4, 4, 4, 4, 4, 4, 4, 4$ $4$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,13, 6,14)( 7,15, 8,16)(17,26,18,25)(19,28,20,27) (21,30,22,29)(23,32,24,31)$ $4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1,11, 2,12)( 3, 9, 4,10)( 5,30, 6,29)( 7,31, 8,32)(13,21,14,22)(15,24,16,23) (17,27,18,28)(19,26,20,25)$ $4, 4, 4, 4, 4, 4, 4, 4$ $4$ $4$ $( 1,13,25, 8)( 2,14,26, 7)( 3,16,28, 6)( 4,15,27, 5)( 9,24,17,30)(10,23,18,29) (11,22,19,31)(12,21,20,32)$ $8, 8, 8, 8$ $2$ $8$ $( 1,15,12,23, 2,16,11,24)( 3,13,10,22, 4,14, 9,21)( 5,20,29,26, 6,19,30,25) ( 7,17,32,28, 8,18,31,27)$ $8, 8, 8, 8$ $2$ $8$ $( 1,16,12,24, 2,15,11,23)( 3,14,10,21, 4,13, 9,22)( 5,19,29,25, 6,20,30,26) ( 7,18,32,27, 8,17,31,28)$ $4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1,19, 2,20)( 3,17, 4,18)( 5,24, 6,23)( 7,22, 8,21)( 9,27,10,28)(11,26,12,25) (13,32,14,31)(15,30,16,29)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,25)( 2,26)( 3,28)( 4,27)( 5,15)( 6,16)( 7,14)( 8,13)( 9,17)(10,18)(11,19) (12,20)(21,32)(22,31)(23,29)(24,30)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,26)( 2,25)( 3,27)( 4,28)( 5,16)( 6,15)( 7,13)( 8,14)( 9,18)(10,17)(11,20) (12,19)(21,31)(22,32)(23,30)(24,29)$

## Group invariants

 Order: $32=2^{5}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [32, 10]
 Character table:  2 5 5 3 4 4 3 3 4 3 4 4 4 5 5 1a 2a 4a 8a 8b 4b 4c 4d 4e 8c 8d 4f 2b 2c 2P 1a 1a 2a 4d 4d 2b 2a 2a 2b 4d 4d 2a 1a 1a 3P 1a 2a 4a 8c 8d 4e 4c 4d 4b 8a 8b 4f 2b 2c 5P 1a 2a 4a 8b 8a 4b 4c 4d 4e 8d 8c 4f 2b 2c 7P 1a 2a 4a 8d 8c 4e 4c 4d 4b 8b 8a 4f 2b 2c 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 A -A 1 -1 A -A -A 1 -1 -1 X.6 1 1 -1 -A -A A 1 -1 -A A A 1 -1 -1 X.7 1 1 1 A A A -1 -1 -A -A -A 1 -1 -1 X.8 1 1 1 -A -A -A -1 -1 A A A 1 -1 -1 X.9 2 2 . . . . . 2 . . . -2 -2 -2 X.10 2 2 . . . . . -2 . . . -2 2 2 X.11 2 -2 . B -B . . . . B -B . 2 -2 X.12 2 -2 . -B B . . . . -B B . 2 -2 X.13 2 -2 . C -C . . . . -C C . -2 2 X.14 2 -2 . -C C . . . . C -C . -2 2 A = -E(4) = -Sqrt(-1) = -i B = -E(8)-E(8)^3 = -Sqrt(-2) = -i2 C = -E(8)+E(8)^3 = -Sqrt(2) = -r2