# Properties

 Label 32T41 Order $$32$$ n $$32$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group Yes Group: $C_2.C_4^2$

## Group action invariants

 Degree $n$ : $32$ Transitive number $t$ : $41$ Group : $C_2.C_4^2$ Parity: $1$ Primitive: No Nilpotency class: $2$ Generators: (1,16,9,24)(2,15,10,23)(3,14,11,22)(4,13,12,21)(5,20,29,28)(6,19,30,27)(7,18,31,26)(8,17,32,25), (1,8,11,30)(2,7,12,29)(3,6,9,32)(4,5,10,31)(13,19,23,25)(14,20,24,26)(15,17,21,27)(16,18,22,28) $|\Aut(F/K)|$: $32$

## 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:  $D_{4}$ x 3, $C_4\times C_2$ x 3, $Q_8$
16:  $C_2^2:C_4$ x 3, $C_4^2$, $C_4:C_4$ x 3

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 6, $C_2^2$, $D_{4}$ x 6

Degree 8: $C_4\times C_2$ x 3, $D_4$ x 3, $Q_8$, $C_2^2:C_4$ x 6

Degree 16: $C_4^2$, $C_4:C_4$ x 3, $C_2^2 : C_4$ x 3

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

## Group invariants

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