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Properties

Label 8T47
8T47 1 2 1->2 5 1->5 3 2->3 2->3 6 2->6 7 3->7 8 3->8 4 4->8 8->1
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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Copy content magma:G := TransitiveGroup(8, 47);
 

Group invariants

Abstract group:  $S_4\wr C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $1152=2^{7} \cdot 3^{2}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$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

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{8}$ $1$ $1$ $0$ $()$
2A $2^{2},1^{4}$ $6$ $2$ $2$ $(1,2)(3,8)$
2B $2^{4}$ $9$ $2$ $4$ $(1,2)(3,8)(4,7)(5,6)$
2C $2,1^{6}$ $12$ $2$ $1$ $(6,7)$
2D $2^{4}$ $24$ $2$ $4$ $(1,7)(2,4)(3,6)(5,8)$
2E $2^{2},1^{4}$ $36$ $2$ $2$ $(1,8)(4,6)$
2F $2^{3},1^{2}$ $36$ $2$ $3$ $(3,8)(4,5)(6,7)$
3A $3,1^{5}$ $16$ $3$ $2$ $(2,8,3)$
3B $3^{2},1^{2}$ $64$ $3$ $4$ $(1,3,2)(4,7,6)$
4A $4,1^{4}$ $12$ $4$ $3$ $(1,3,2,8)$
4B $4^{2}$ $36$ $4$ $6$ $(1,8,2,3)(4,5,6,7)$
4C $4,2^{2}$ $36$ $4$ $5$ $(1,8)(2,3)(4,5,6,7)$
4D $4,2,1^{2}$ $72$ $4$ $4$ $(1,8)(4,5,7,6)$
4E $4^{2}$ $72$ $4$ $6$ $(1,7,2,4)(3,6,8,5)$
4F $4,2^{2}$ $144$ $4$ $5$ $(1,4,8,6)(2,7)(3,5)$
6A $3,2^{2},1$ $48$ $6$ $4$ $(1,2)(3,8)(5,7,6)$
6B $3,2,1^{3}$ $96$ $6$ $3$ $(2,3,8)(6,7)$
6C $6,2$ $192$ $6$ $6$ $(1,4,3,7,2,6)(5,8)$
8A $8$ $144$ $8$ $7$ $(1,4,8,5,2,6,3,7)$
12A $4,3,1$ $96$ $12$ $5$ $(1,8,2,3)(5,6,7)$

Malle's constant $a(G)$:     $1$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 3A 3B 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A 12A
Size 1 6 9 12 24 36 36 16 64 12 36 36 72 72 144 48 96 192 144 96
2 P 1A 1A 1A 1A 1A 1A 1A 3A 3B 2A 2B 2A 2A 2B 2E 3A 3A 3B 4B 6A
3 P 1A 2A 2B 2C 2D 2E 2F 1A 1A 4A 4B 4C 4D 4E 4F 2A 2C 2D 8A 4A
Type
1152.157849.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1152.157849.1b R 1 1 1 −1 −1 1 −1 1 1 −1 1 −1 1 −1 1 1 −1 −1 1 −1
1152.157849.1c R 1 1 1 −1 1 1 −1 1 1 −1 1 −1 1 1 −1 1 −1 1 −1 −1
1152.157849.1d R 1 1 1 1 −1 1 1 1 1 1 1 1 1 −1 −1 1 1 −1 −1 1
1152.157849.2a R 2 2 2 0 0 −2 0 2 2 0 −2 0 −2 0 0 2 0 0 0 0
1152.157849.4a R 4 4 4 0 2 0 0 −2 1 0 0 0 0 2 0 −2 0 −1 0 0
1152.157849.4b R 4 4 4 2 0 0 2 1 −2 2 0 2 0 0 0 1 −1 0 0 −1
1152.157849.4c R 4 4 4 −2 0 0 −2 1 −2 −2 0 −2 0 0 0 1 1 0 0 1
1152.157849.4d R 4 4 4 0 −2 0 0 −2 1 0 0 0 0 −2 0 −2 0 1 0 0
1152.157849.6a R 6 2 −2 4 0 2 0 3 0 2 −2 −2 0 0 0 −1 1 0 0 −1
1152.157849.6b R 6 2 −2 2 0 −2 −2 3 0 4 2 0 0 0 0 −1 −1 0 0 1
1152.157849.6c R 6 2 −2 −2 0 −2 2 3 0 −4 2 0 0 0 0 −1 1 0 0 −1
1152.157849.6d R 6 2 −2 −4 0 2 0 3 0 −2 −2 2 0 0 0 −1 −1 0 0 1
1152.157849.9a R 9 −3 1 3 3 1 −1 0 0 −3 1 1 −1 −1 1 0 0 0 −1 0
1152.157849.9b R 9 −3 1 −3 3 1 1 0 0 3 1 −1 −1 −1 −1 0 0 0 1 0
1152.157849.9c R 9 −3 1 −3 −3 1 1 0 0 3 1 −1 −1 1 1 0 0 0 −1 0
1152.157849.9d R 9 −3 1 3 −3 1 −1 0 0 −3 1 1 −1 1 −1 0 0 0 1 0
1152.157849.12a R 12 4 −4 2 0 0 2 −3 0 −2 0 −2 0 0 0 1 −1 0 0 1
1152.157849.12b R 12 4 −4 −2 0 0 −2 −3 0 2 0 2 0 0 0 1 1 0 0 −1
1152.157849.18a R 18 −6 2 0 0 −2 0 0 0 0 −2 0 2 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{8} + 2 x^{5} + x^{2} + t$ Copy content Toggle raw display