# Properties

 Label 32T26 Order $$32$$ n $$32$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $D_8:C_2$

## Group action invariants

 Degree $n$ : $32$ Transitive number $t$ : $26$ Group : $D_8:C_2$ Parity: $1$ Primitive: No Nilpotency class: $3$ Generators: (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,19)(14,20)(15,18)(16,17)(21,26)(22,25)(23,27)(24,28)(29,32)(30,31), (1,27,29,24,4,26,31,22)(2,28,30,23,3,25,32,21)(5,14,10,19,7,16,11,18)(6,13,9,20,8,15,12,17), (1,16)(2,15)(3,13)(4,14)(5,27)(6,28)(7,26)(8,25)(9,21)(10,22)(11,24)(12,23)(17,32)(18,31)(19,29)(20,30) $|\Aut(F/K)|$: $32$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$
16:  $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_2^3$, $D_4$ x 2, $D_4\times C_2$ x 4

Degree 16: $D_4\times C_2$, 16T44 x 2, 16T47

## Low degree siblings

16T44 x 2, 16T47

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [32, 42]
 Character table:  2 5 3 5 3 5 5 4 4 3 4 3 4 4 4 1a 2a 2b 4a 4b 4c 2c 8a 2d 8b 4d 8c 8d 4e 2P 1a 1a 1a 2b 2b 2b 1a 4e 1a 4e 2b 4e 4e 2b 3P 1a 2a 2b 4a 4c 4b 2c 8b 2d 8a 4d 8c 8d 4e 5P 1a 2a 2b 4a 4b 4c 2c 8b 2d 8a 4d 8d 8c 4e 7P 1a 2a 2b 4a 4c 4b 2c 8a 2d 8b 4d 8d 8c 4e 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 . A -A . B . -B . C -C . X.12 2 . -2 . A -A . -B . B . -C C . X.13 2 . -2 . -A A . B . -B . -C C . X.14 2 . -2 . -A A . -B . B . C -C . A = -2*E(4) = -2*Sqrt(-1) = -2i B = -E(8)+E(8)^3 = -Sqrt(2) = -r2 C = E(8)+E(8)^3 = Sqrt(-2) = i2