# Properties

 Label 32T24 Order $$32$$ n $$32$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_2^2\wr C_2$

## Group action invariants

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

## 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 6, $C_2^3$
16:  $D_4\times C_2$ x 3

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 12

Degree 8: $C_2^3$, $D_4$ x 6, $D_4\times C_2$ x 12, $C_2^2 \wr C_2$ x 8

Degree 16: $D_4\times C_2$ x 3, 16T39 x 6, 16T46

## Low degree siblings

8T18 x 8, 16T39 x 6, 16T46

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [32, 27]
 Character table:  2 5 4 4 4 4 5 4 5 4 3 3 3 3 5 1a 2a 2b 2c 2d 2e 2f 2g 2h 4a 2i 4b 4c 2j 2P 1a 1a 1a 1a 1a 1a 1a 1a 1a 2g 1a 2e 2j 1a 3P 1a 2a 2b 2c 2d 2e 2f 2g 2h 4a 2i 4b 4c 2j 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 . -2 . . . . . 2 X.12 2 . . -2 . 2 . -2 2 . . . . -2 X.13 2 . . 2 . 2 . -2 -2 . . . . -2 X.14 2 . 2 . -2 -2 . -2 . . . . . 2