Properties

Label 32T45
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_4:Q_8$

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Group action invariants

Degree $n$ :  $32$
Transitive number $t$ :  $45$
Group :  $C_4:Q_8$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,30,3,32)(2,29,4,31)(5,10,7,12)(6,9,8,11)(13,20,15,18)(14,19,16,17)(21,28,23,26)(22,27,24,25), (1,22,3,24)(2,21,4,23)(5,20,7,18)(6,19,8,17)(9,14,11,16)(10,13,12,15)(25,32,27,30)(26,31,28,29), (1,8,3,6)(2,7,4,5)(9,29,11,31)(10,30,12,32)(13,27,15,25)(14,28,16,26)(17,22,19,24)(18,21,20,23)
$|\Aut(F/K)|$:  $32$

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, $C_2^3$, $Q_8$ x 4
16:  $D_4\times C_2$, $D_8$ x 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, $Q_8$ x 4, $D_4\times C_2$ x 4

Degree 16: $D_8$ x 2, $D_4\times C_2$

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 TypeSizeOrderRepresentative
$ 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 $ $2$ $4$ $( 1,13, 4,16)( 2,14, 3,15)( 5,27, 8,26)( 6,28, 7,25)( 9,22,12,23)(10,21,11,24) (17,31,20,30)(18,32,19,29)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,14, 4,15)( 2,13, 3,16)( 5,28, 8,25)( 6,27, 7,26)( 9,21,12,24)(10,22,11,23) (17,32,20,29)(18,31,19,30)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1,17, 3,19)( 2,18, 4,20)( 5,21, 7,23)( 6,22, 8,24)( 9,26,11,28)(10,25,12,27) (13,30,15,32)(14,29,16,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,21, 3,23)( 2,22, 4,24)( 5,19, 7,17)( 6,20, 8,18)( 9,13,11,15)(10,14,12,16) (25,31,27,29)(26,32,28,30)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,22, 3,24)( 2,21, 4,23)( 5,20, 7,18)( 6,19, 8,17)( 9,14,11,16)(10,13,12,15) (25,32,27,30)(26,31,28,29)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1,25, 3,27)( 2,26, 4,28)( 5,13, 7,15)( 6,14, 8,16)( 9,17,11,19)(10,18,12,20) (21,29,23,31)(22,30,24,32)$
$ 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, 35]
Character table:   
      2  5  5  5  5  3  4  4  4  4  3  4  4  3  3

        1a 2a 2b 2c 4a 4b 4c 4d 4e 4f 4g 4h 4i 4j
     2P 1a 1a 1a 1a 2b 2a 2a 2c 2c 2b 2b 2b 2b 2b
     3P 1a 2a 2b 2c 4a 4b 4c 4d 4e 4f 4g 4h 4i 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  .  .  . -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  .  .