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Properties

Label 32T28
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_8.C_4$

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Group action invariants

Degree $n$:  $32$
Transitive number $t$:  $28$
Group:  $C_8.C_4$
Parity:  $1$
Primitive:  no
Nilpotency class:  $3$
$|\Aut(F/K)|$:  $32$
Generators:  (1,23,12,30,3,21,10,32)(2,24,11,29,4,22,9,31)(5,19,13,28,8,17,15,25)(6,20,14,27,7,18,16,26), (1,18,10,27,3,20,12,26)(2,17,9,28,4,19,11,25)(5,32,15,21,8,30,13,23)(6,31,16,22,7,29,14,24)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$, $C_4\times C_2$, $Q_8$
$16$:  $C_4:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $C_4\times C_2$, $D_4$, $Q_8$

Degree 16: $C_4:C_4$, 16T49

Low degree siblings

16T49

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [32, 15]
Character table:    
      2  5  4  5  4  4  4  4  4  5  5  3  3  3  3

        1a 2a 2b 8a 8b 8c 8d 4a 4b 4c 8e 8f 8g 8h
     2P 1a 1a 1a 4a 4a 4a 4a 2b 2b 2b 4c 4b 4c 4b
     3P 1a 2a 2b 8d 8b 8c 8a 4a 4c 4b 8f 8e 8h 8g
     5P 1a 2a 2b 8d 8c 8b 8a 4a 4b 4c 8e 8f 8g 8h
     7P 1a 2a 2b 8a 8c 8b 8d 4a 4c 4b 8f 8e 8h 8g

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  D -D -D  D
X.6      1 -1  1 -1  1  1 -1  1 -1 -1 -D  D  D -D
X.7      1 -1  1  1 -1 -1  1  1 -1 -1  D -D  D -D
X.8      1 -1  1  1 -1 -1  1  1 -1 -1 -D  D -D  D
X.9      2  2  2  .  .  .  . -2 -2 -2  .  .  .  .
X.10     2 -2  2  .  .  .  . -2  2  2  .  .  .  .
X.11     2  . -2  A  B -B -A  .  C -C  .  .  .  .
X.12     2  . -2  A -B  B -A  . -C  C  .  .  .  .
X.13     2  . -2 -A  B -B  A  . -C  C  .  .  .  .
X.14     2  . -2 -A -B  B  A  .  C -C  .  .  .  .

A = -E(8)+E(8)^3
  = -Sqrt(2) = -r2
B = -E(8)-E(8)^3
  = -Sqrt(-2) = -i2
C = -2*E(4)
  = -2*Sqrt(-1) = -2i
D = -E(4)
  = -Sqrt(-1) = -i