# Properties

 Label 32T46 Order $$32$$ n $$32$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_4.Q_8$

## Group action invariants

 Degree $n$ : $32$ Transitive number $t$ : $46$ Group : $C_4.Q_8$ Parity: $1$ Primitive: No Nilpotency class: $2$ 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) $|\Aut(F/K)|$: $32$

## 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