# Properties

 Label 32T42 Order $$32$$ n $$32$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_2.OD_{16}$

## Group action invariants

 Degree $n$ : $32$ Transitive number $t$ : $42$ Group : $C_2.OD_{16}$ Parity: $1$ Primitive: No Nilpotency class: $2$ Generators: (1,6,11,31)(2,5,12,32)(3,7,9,29)(4,8,10,30)(13,17,23,28)(14,18,24,27)(15,20,21,26)(16,19,22,25), (1,23,10,16,2,24,9,15)(3,21,11,13,4,22,12,14)(5,28,29,19,6,27,30,20)(7,25,31,18,8,26,32,17) $|\Aut(F/K)|$: $32$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 6, $C_2^2$
8:  $C_4\times C_2$ x 3
16:  $C_8:C_2$ x 2, $C_4^2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 6, $C_2^2$

Degree 8: $C_4\times C_2$ x 3, $C_8:C_2$ x 2

Degree 16: $C_4^2$, $C_8: C_2$ 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)$ $4, 4, 4, 4, 4, 4, 4, 4$ $1$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)(17,20,18,19)(21,24,22,23) (25,28,26,27)(29,32,30,31)$ $4, 4, 4, 4, 4, 4, 4, 4$ $1$ $4$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)(17,19,18,20)(21,23,22,24) (25,27,26,28)(29,31,30,32)$ $4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1, 5,11,32)( 2, 6,12,31)( 3, 8, 9,30)( 4, 7,10,29)(13,18,23,27)(14,17,24,28) (15,19,21,25)(16,20,22,26)$ $4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1, 7,12,30)( 2, 8,11,29)( 3, 5,10,31)( 4, 6, 9,32)(13,20,24,25)(14,19,23,26) (15,18,22,28)(16,17,21,27)$ $4, 4, 4, 4, 4, 4, 4, 4$ $1$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,30, 6,29)( 7,32, 8,31)(13,21,14,22)(15,24,16,23) (17,26,18,25)(19,28,20,27)$ $4, 4, 4, 4, 4, 4, 4, 4$ $1$ $4$ $( 1,10, 2, 9)( 3,11, 4,12)( 5,29, 6,30)( 7,31, 8,32)(13,22,14,21)(15,23,16,24) (17,25,18,26)(19,27,20,28)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 5,32)( 6,31)( 7,29)( 8,30)(13,23)(14,24)(15,21) (16,22)(17,28)(18,27)(19,25)(20,26)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,12)( 2,11)( 3,10)( 4, 9)( 5,31)( 6,32)( 7,30)( 8,29)(13,24)(14,23)(15,22) (16,21)(17,27)(18,28)(19,26)(20,25)$ $8, 8, 8, 8$ $2$ $8$ $( 1,13,10,22, 2,14, 9,21)( 3,15,11,23, 4,16,12,24)( 5,17,29,25, 6,18,30,26) ( 7,19,31,27, 8,20,32,28)$ $8, 8, 8, 8$ $2$ $8$ $( 1,15, 9,24, 2,16,10,23)( 3,14,12,22, 4,13,11,21)( 5,20,30,27, 6,19,29,28) ( 7,17,32,26, 8,18,31,25)$ $8, 8, 8, 8$ $2$ $8$ $( 1,17, 3,20, 2,18, 4,19)( 5,23, 8,21, 6,24, 7,22)( 9,26,12,27,10,25,11,28) (13,30,15,31,14,29,16,32)$ $8, 8, 8, 8$ $2$ $8$ $( 1,19, 4,18, 2,20, 3,17)( 5,22, 7,24, 6,21, 8,23)( 9,28,11,25,10,27,12,26) (13,32,16,29,14,31,15,30)$ $8, 8, 8, 8$ $2$ $8$ $( 1,21, 9,14, 2,22,10,13)( 3,24,12,16, 4,23,11,15)( 5,26,30,18, 6,25,29,17) ( 7,28,32,20, 8,27,31,19)$ $8, 8, 8, 8$ $2$ $8$ $( 1,23,10,16, 2,24, 9,15)( 3,21,11,13, 4,22,12,14)( 5,28,29,19, 6,27,30,20) ( 7,25,31,18, 8,26,32,17)$ $8, 8, 8, 8$ $2$ $8$ $( 1,25, 4,27, 2,26, 3,28)( 5,16, 7,14, 6,15, 8,13)( 9,17,11,19,10,18,12,20) (21,30,23,32,22,29,24,31)$ $8, 8, 8, 8$ $2$ $8$ $( 1,27, 3,25, 2,28, 4,26)( 5,14, 8,16, 6,13, 7,15)( 9,19,12,17,10,20,11,18) (21,32,24,30,22,31,23,29)$ $4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1,29,12, 8)( 2,30,11, 7)( 3,32,10, 6)( 4,31, 9, 5)(13,26,24,19)(14,25,23,20) (15,27,22,17)(16,28,21,18)$ $4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1,31,11, 6)( 2,32,12, 5)( 3,29, 9, 7)( 4,30,10, 8)(13,28,23,17)(14,27,24,18) (15,26,21,20)(16,25,22,19)$

## Group invariants

 Order: $32=2^{5}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [32, 4]
 Character table:  2 5 5 5 5 4 4 5 5 5 5 4 4 4 4 4 4 4 4 4 4 1a 2a 4a 4b 4c 4d 4e 4f 2b 2c 8a 8b 8c 8d 8e 8f 8g 8h 4g 4h 2P 1a 1a 2a 2a 2b 2c 2a 2a 1a 1a 4f 4e 4a 4b 4e 4f 4b 4a 2c 2b 3P 1a 2a 4b 4a 4h 4g 4f 4e 2b 2c 8e 8f 8d 8c 8a 8b 8h 8g 4d 4c 5P 1a 2a 4a 4b 4c 4d 4e 4f 2b 2c 8a 8b 8c 8d 8e 8f 8g 8h 4g 4h 7P 1a 2a 4b 4a 4h 4g 4f 4e 2b 2c 8e 8f 8d 8c 8a 8b 8h 8g 4d 4c X.1 1 1 1 1 1 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 -1 -1 1 1 -1 -1 X.3 1 1 1 1 -1 -1 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 -1 -1 -1 -1 1 1 X.5 1 1 -1 -1 -1 1 -1 -1 1 1 B -B -B B -B B B -B 1 -1 X.6 1 1 -1 -1 -1 1 -1 -1 1 1 -B B B -B B -B -B B 1 -1 X.7 1 1 -1 -1 1 -1 -1 -1 1 1 B -B B -B -B B -B B -1 1 X.8 1 1 -1 -1 1 -1 -1 -1 1 1 -B B -B B B -B B -B -1 1 X.9 1 1 -1 -1 B -B 1 1 -1 -1 -1 1 -B B -1 1 -B B B -B X.10 1 1 -1 -1 -B B 1 1 -1 -1 -1 1 B -B -1 1 B -B -B B X.11 1 1 -1 -1 B -B 1 1 -1 -1 1 -1 B -B 1 -1 B -B B -B X.12 1 1 -1 -1 -B B 1 1 -1 -1 1 -1 -B B 1 -1 -B B -B B X.13 1 1 1 1 B B -1 -1 -1 -1 B B -1 -1 -B -B 1 1 -B -B X.14 1 1 1 1 -B -B -1 -1 -1 -1 -B -B -1 -1 B B 1 1 B B X.15 1 1 1 1 B B -1 -1 -1 -1 -B -B 1 1 B B -1 -1 -B -B X.16 1 1 1 1 -B -B -1 -1 -1 -1 B B 1 1 -B -B -1 -1 B B X.17 2 -2 A -A . . A -A 2 -2 . . . . . . . . . . X.18 2 -2 -A A . . -A A 2 -2 . . . . . . . . . . X.19 2 -2 A -A . . -A A -2 2 . . . . . . . . . . X.20 2 -2 -A A . . A -A -2 2 . . . . . . . . . . A = -2*E(4) = -2*Sqrt(-1) = -2i B = -E(4) = -Sqrt(-1) = -i