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Properties

Label 8T44
Order \(384\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2 \wr S_4$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $44$
Group :  $C_2 \wr S_4$
CHM label :  $[2^{4}]S(4)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,3,8)(4,5,6,7), (1,8)(4,5), (4,8)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
12:  $D_{6}$
24:  $S_4$
48:  $S_4\times C_2$
192:  $V_4^2:(S_3\times C_2)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $S_4$

Low degree siblings

8T44 x 3, 16T736 x 2, 16T743 x 2, 16T748 x 2, 16T752 x 2, 24T708 x 4, 24T1151 x 4, 32T9340, 32T9355, 32T9459 x 4

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 $ $1$ $1$ $()$
$ 2, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $(4,8)$
$ 2, 2, 1, 1, 1, 1 $ $12$ $2$ $(3,4)(7,8)$
$ 4, 1, 1, 1, 1 $ $12$ $4$ $(3,4,7,8)$
$ 2, 2, 1, 1, 1, 1 $ $6$ $2$ $(3,7)(4,8)$
$ 2, 2, 2, 1, 1 $ $24$ $2$ $(2,3)(4,8)(6,7)$
$ 3, 3, 1, 1 $ $32$ $3$ $(2,3,4)(6,7,8)$
$ 6, 1, 1 $ $32$ $6$ $(2,3,4,6,7,8)$
$ 4, 2, 1, 1 $ $24$ $4$ $(2,3,6,7)(4,8)$
$ 2, 2, 2, 1, 1 $ $4$ $2$ $(2,6)(3,7)(4,8)$
$ 2, 2, 2, 2 $ $12$ $2$ $(1,2)(3,4)(5,6)(7,8)$
$ 4, 2, 2 $ $24$ $4$ $(1,2)(3,4,7,8)(5,6)$
$ 2, 2, 2, 2 $ $12$ $2$ $(1,2)(3,7)(4,8)(5,6)$
$ 3, 3, 2 $ $32$ $6$ $(1,2,3)(4,8)(5,6,7)$
$ 4, 4 $ $48$ $4$ $(1,2,3,4)(5,6,7,8)$
$ 8 $ $48$ $8$ $(1,2,3,4,5,6,7,8)$
$ 6, 2 $ $32$ $6$ $(1,2,3,5,6,7)(4,8)$
$ 4, 4 $ $12$ $4$ $(1,2,5,6)(3,4,7,8)$
$ 4, 2, 2 $ $12$ $4$ $(1,2,5,6)(3,7)(4,8)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$

Group invariants

Order:  $384=2^{7} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [384, 5602]
Character table:    
      2  7  5  5  5  6  4  2  2  4  5  5  4  5  2  3  3  2  5  5  7
      3  1  1  .  .  .  .  1  1  .  1  .  .  .  1  .  .  1  .  .  1

        1a 2a 2b 4a 2c 2d 3a 6a 4b 2e 2f 4c 2g 6b 4d 8a 6c 4e 4f 2h
     2P 1a 1a 1a 2c 1a 1a 3a 3a 2c 1a 1a 2c 1a 3a 2f 4e 3a 2h 2c 1a
     3P 1a 2a 2b 4a 2c 2d 1a 2e 4b 2e 2f 4c 2g 2a 4d 8a 2h 4e 4f 2h
     5P 1a 2a 2b 4a 2c 2d 3a 6a 4b 2e 2f 4c 2g 6b 4d 8a 6c 4e 4f 2h
     7P 1a 2a 2b 4a 2c 2d 3a 6a 4b 2e 2f 4c 2g 6b 4d 8a 6c 4e 4f 2h

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      2 -2  .  .  2  . -1  1  . -2  2 -2  .  1  .  . -1  2  .  2
X.6      2  2  .  .  2  . -1 -1  .  2  2  2  . -1  .  . -1  2  .  2
X.7      3 -3 -1  1  3  1  .  . -1 -3 -1  1 -1  .  1 -1  . -1  1  3
X.8      3 -3  1 -1  3 -1  .  .  1 -3 -1  1  1  . -1  1  . -1 -1  3
X.9      3  3 -1 -1  3 -1  .  . -1  3 -1 -1 -1  .  1  1  . -1 -1  3
X.10     3  3  1  1  3  1  .  .  1  3 -1 -1  1  . -1 -1  . -1  1  3
X.11     4 -2  2 -2  .  .  1 -1  .  2  .  . -2  1  .  . -1  .  2 -4
X.12     4 -2 -2  2  .  .  1 -1  .  2  .  .  2  1  .  . -1  . -2 -4
X.13     4  2  2  2  .  .  1  1  . -2  .  . -2 -1  .  . -1  . -2 -4
X.14     4  2 -2 -2  .  .  1  1  . -2  .  .  2 -1  .  . -1  .  2 -4
X.15     6  . -2  . -2  .  .  .  2  .  2  . -2  .  .  .  . -2  .  6
X.16     6  .  2  . -2  .  .  . -2  .  2  .  2  .  .  .  . -2  .  6
X.17     6  .  . -2 -2  2  .  .  .  . -2  .  .  .  .  .  .  2 -2  6
X.18     6  .  .  2 -2 -2  .  .  .  . -2  .  .  .  .  .  .  2  2  6
X.19     8 -4  .  .  .  . -1  1  .  4  .  .  . -1  .  .  1  .  . -8
X.20     8  4  .  .  .  . -1 -1  . -4  .  .  .  1  .  .  1  .  . -8