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Properties

Label 8T47
Degree $8$
Order $1152$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_4\wr C_2$

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Show commands: Magma

magma: G := TransitiveGroup(8, 47);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $47$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_4\wr C_2$
CHM label:  $[S(4)^{2}]2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,3,8), (2,3), (1,5)(2,6)(3,7)(4,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$
$72$:  $C_3^2:D_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Low degree siblings

12T200, 12T201, 12T202, 12T203, 16T1292, 16T1294, 16T1295, 16T1296, 18T272, 18T273, 18T274, 18T275, 24T2803, 24T2804, 24T2805, 24T2806, 24T2807, 24T2808, 24T2809, 24T2810, 24T2821, 24T2826, 32T96692, 32T96694, 32T96695, 32T96696, 36T1758, 36T1759, 36T1760, 36T1761, 36T1762, 36T1763, 36T1764, 36T1765, 36T1766, 36T1767, 36T1768, 36T1769, 36T1943, 36T1944, 36T1945, 36T1946

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, 2, 1, 1, 1, 1 $ $6$ $2$ $(1,3)(2,8)$
$ 2, 2, 2, 2 $ $9$ $2$ $(1,3)(2,8)(4,6)(5,7)$
$ 3, 1, 1, 1, 1, 1 $ $16$ $3$ $(2,3,8)$
$ 3, 2, 2, 1 $ $48$ $6$ $(2,3,8)(4,6)(5,7)$
$ 3, 3, 1, 1 $ $64$ $3$ $(2,3,8)(5,7,6)$
$ 2, 1, 1, 1, 1, 1, 1 $ $12$ $2$ $(3,8)$
$ 4, 1, 1, 1, 1 $ $12$ $4$ $(1,3,2,8)$
$ 2, 2, 2, 1, 1 $ $36$ $2$ $(3,8)(4,6)(5,7)$
$ 4, 2, 2 $ $36$ $4$ $(1,3,2,8)(4,6)(5,7)$
$ 3, 2, 1, 1, 1 $ $96$ $6$ $(3,8)(5,7,6)$
$ 4, 3, 1 $ $96$ $12$ $(1,3,2,8)(5,7,6)$
$ 2, 2, 1, 1, 1, 1 $ $36$ $2$ $(3,8)(6,7)$
$ 4, 2, 1, 1 $ $72$ $4$ $(1,3,2,8)(6,7)$
$ 4, 4 $ $36$ $4$ $(1,3,2,8)(4,6,5,7)$
$ 2, 2, 2, 2 $ $24$ $2$ $(1,5)(2,6)(3,7)(4,8)$
$ 4, 4 $ $72$ $4$ $(1,5,3,7)(2,6,8,4)$
$ 6, 2 $ $192$ $6$ $(1,5)(2,6,3,7,8,4)$
$ 4, 2, 2 $ $144$ $4$ $(1,5)(2,6)(3,7,8,4)$
$ 8 $ $144$ $8$ $(1,5,3,7,2,6,8,4)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1152=2^{7} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  1152.157849
magma: IdentifyGroup(G);
 
Character table:    
      2  7  6  7  3  3  1  5  5  5  5  2   2  5  4  5  4  4  1  3  3
      3  2  1  .  2  1  2  1  1  .  .  1   1  .  .  .  1  .  1  .  .

        1a 2a 2b 3a 6a 3b 2c 4a 2d 4b 6b 12a 2e 4c 4d 2f 4e 6c 4f 8a
     2P 1a 1a 1a 3a 3a 3b 1a 2a 1a 2a 3a  6a 1a 2a 2b 1a 2b 3b 2e 4d
     3P 1a 2a 2b 1a 2a 1a 2c 4a 2d 4b 2c  4a 2e 4c 4d 2f 4e 2f 4f 8a
     5P 1a 2a 2b 3a 6a 3b 2c 4a 2d 4b 6b 12a 2e 4c 4d 2f 4e 6c 4f 8a
     7P 1a 2a 2b 3a 6a 3b 2c 4a 2d 4b 6b 12a 2e 4c 4d 2f 4e 6c 4f 8a
    11P 1a 2a 2b 3a 6a 3b 2c 4a 2d 4b 6b 12a 2e 4c 4d 2f 4e 6c 4f 8a

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

magma: CharacterTable(G);