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Properties

Label 12T137
12T137 1 5 1->5 12 1->12 1->12 2 3 2->3 6 2->6 8 2->8 10 2->10 7 3->7 9 3->9 11 3->11 4 4->5 4->8 4->8 4->10 5->9 5->9 5->11 6->7 6->10 7->11 8->12 9->1 10->2 11->3 12->4
Degree $12$
Order $384$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^4:S_4$

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Copy content magma:G := TransitiveGroup(12, 137);
 

Group invariants

Abstract group:  $C_2^4:S_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $384=2^{7} \cdot 3$
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 $n$:  $12$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $137$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[2^{4}]S_{4}(6d)$
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(2,8)(3,9)(4,10)(5,11)$, $(1,12)(2,3)(4,5)$, $(1,5,9)(2,6,10)(3,7,11)(4,8,12)$, $(1,12)(6,7)$, $(2,10)(3,11)(4,8)(5,9)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$
$8$:  $C_2^3$
$12$:  $D_{6}$ x 3
$24$:  $S_4$, $S_3 \times C_2^2$
$48$:  $S_4\times C_2$ x 3
$96$:  12T48
$192$:  $V_4^2:(S_3\times C_2)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4$

Low degree siblings

12T136 x 4, 12T137 x 3, 16T724 x 4, 24T1143 x 2, 24T1144 x 2, 24T1184 x 2, 24T1185 x 4, 24T1186 x 2, 24T1187 x 4, 24T1188 x 4, 24T1189 x 4, 24T1190 x 4, 24T1191 x 2, 24T1192 x 4, 24T1193 x 2, 24T1194 x 4, 24T1195 x 8, 24T1196 x 8, 24T1197 x 8, 24T1198 x 8, 32T9332 x 2, 32T9457 x 4

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{12}$ $1$ $1$ $0$ $()$
2A $2^{6}$ $1$ $2$ $6$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$
2B $2^{2},1^{8}$ $3$ $2$ $2$ $( 1,12)( 6, 7)$
2C $2^{4},1^{4}$ $3$ $2$ $4$ $( 2, 3)( 4, 5)( 8, 9)(10,11)$
2D $2^{3},1^{6}$ $4$ $2$ $3$ $(2,3)(4,5)(6,7)$
2E $2^{3},1^{6}$ $4$ $2$ $3$ $( 1,12)( 2, 3)( 4, 5)$
2F $2^{4},1^{4}$ $6$ $2$ $4$ $( 1, 6)( 4,10)( 5,11)( 7,12)$
2G $2^{6}$ $6$ $2$ $6$ $( 1, 7)( 2, 8)( 3, 9)( 4, 5)( 6,12)(10,11)$
2H $2^{4},1^{4}$ $6$ $2$ $4$ $( 2, 9)( 3, 8)( 4,11)( 5,10)$
2I $2^{6}$ $6$ $2$ $6$ $( 1, 7)( 2, 9)( 3, 8)( 4, 5)( 6,12)(10,11)$
2J $2^{5},1^{2}$ $12$ $2$ $5$ $( 1,12)( 2, 5)( 3, 4)( 8,10)( 9,11)$
2K $2^{5},1^{2}$ $12$ $2$ $5$ $( 2, 4)( 3, 5)( 6, 7)( 8,11)( 9,10)$
2L $2^{4},1^{4}$ $12$ $2$ $4$ $( 1, 3)( 2,12)( 6, 8)( 7, 9)$
2M $2^{6}$ $12$ $2$ $6$ $( 1, 2)( 3,12)( 4, 5)( 6, 9)( 7, 8)(10,11)$
3A $3^{4}$ $32$ $3$ $8$ $( 1, 8,11)( 2, 5, 7)( 3, 4, 6)( 9,10,12)$
4A $4^{2},2,1^{2}$ $12$ $4$ $7$ $( 2, 9, 3, 8)( 4,10, 5,11)( 6, 7)$
4B $4^{2},2,1^{2}$ $12$ $4$ $7$ $( 1,12)( 2, 8, 3, 9)( 4,11, 5,10)$
4C $4^{2},2^{2}$ $12$ $4$ $8$ $( 1,12)( 2,11, 3,10)( 4, 8, 5, 9)( 6, 7)$
4D $4^{2},1^{4}$ $12$ $4$ $6$ $( 2,10, 3,11)( 4, 9, 5, 8)$
4E $4^{2},2,1^{2}$ $12$ $4$ $7$ $( 1, 9,12, 8)( 2, 6, 3, 7)( 4, 5)$
4F $4^{2},2,1^{2}$ $12$ $4$ $7$ $( 1, 8,12, 9)( 2, 7, 3, 6)(10,11)$
4G $4^{2},2^{2}$ $24$ $4$ $8$ $( 1, 7)( 2, 5, 9,10)( 3, 4, 8,11)( 6,12)$
4H $4^{2},2^{2}$ $24$ $4$ $8$ $( 1, 7)( 2,10, 8, 4)( 3,11, 9, 5)( 6,12)$
4I $4^{3}$ $24$ $4$ $9$ $( 1, 8, 7, 3)( 2,12, 9, 6)( 4,10, 5,11)$
4J $4^{3}$ $24$ $4$ $9$ $( 1, 3, 6, 9)( 2, 7, 8,12)( 4,11, 5,10)$
6A $6,3^{2}$ $32$ $6$ $9$ $( 1,11, 8)( 2, 6, 5, 3, 7, 4)( 9,12,10)$
6B $6,3^{2}$ $32$ $6$ $9$ $( 1, 4, 3,12, 5, 2)( 6,10, 8)( 7,11, 9)$
6C $6^{2}$ $32$ $6$ $10$ $( 1,11, 8,12,10, 9)( 2, 6, 4, 3, 7, 5)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 3A 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C
Size 1 1 3 3 4 4 6 6 6 6 12 12 12 12 32 12 12 12 12 12 12 24 24 24 24 32 32 32
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 3A 2C 2C 2C 2C 2C 2C 2H 2H 2I 2I 3A 3A 3A
3 P 1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 1A 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 2D 2E 2A
Type
384.17948.1a R 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
384.17948.1b R 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
384.17948.1c R 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
384.17948.1d R 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
384.17948.1e R 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
384.17948.1f R 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
384.17948.1g R 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
384.17948.1h R 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
384.17948.2a R 2 2 2 2 2 2 2 2 2 2 0 0 0 0 −1 0 0 2 2 0 0 0 0 0 0 −1 −1 −1
384.17948.2b R 2 −2 −2 2 −2 2 2 −2 −2 2 0 0 0 0 −1 0 0 −2 2 0 0 0 0 0 0 −1 1 1
384.17948.2c R 2 −2 −2 2 2 −2 2 −2 −2 2 0 0 0 0 −1 0 0 2 −2 0 0 0 0 0 0 1 1 −1
384.17948.2d R 2 2 2 2 −2 −2 2 2 2 2 0 0 0 0 −1 0 0 −2 −2 0 0 0 0 0 0 1 −1 1
384.17948.3a R 3 3 3 3 3 3 −1 −1 −1 −1 1 1 1 1 0 1 1 −1 −1 1 1 −1 −1 −1 −1 0 0 0
384.17948.3b R 3 3 3 3 3 3 −1 −1 −1 −1 −1 −1 −1 −1 0 −1 −1 −1 −1 −1 −1 1 1 1 1 0 0 0
384.17948.3c R 3 −3 −3 3 −3 3 −1 1 1 −1 −1 1 −1 1 0 −1 1 1 −1 1 −1 1 −1 −1 1 0 0 0
384.17948.3d R 3 −3 −3 3 −3 3 −1 1 1 −1 1 −1 1 −1 0 1 −1 1 −1 −1 1 −1 1 1 −1 0 0 0
384.17948.3e R 3 −3 −3 3 3 −3 −1 1 1 −1 −1 1 1 −1 0 −1 1 −1 1 −1 1 −1 1 −1 1 0 0 0
384.17948.3f R 3 −3 −3 3 3 −3 −1 1 1 −1 1 −1 −1 1 0 1 −1 −1 1 1 −1 1 −1 1 −1 0 0 0
384.17948.3g R 3 3 3 3 −3 −3 −1 −1 −1 −1 −1 −1 1 1 0 −1 −1 1 1 1 1 −1 −1 1 1 0 0 0
384.17948.3h R 3 3 3 3 −3 −3 −1 −1 −1 −1 1 1 −1 −1 0 1 1 1 1 −1 −1 1 1 −1 −1 0 0 0
384.17948.6a R 6 6 −2 −2 0 0 −2 2 −2 2 0 0 2 2 0 0 0 0 0 −2 −2 0 0 0 0 0 0 0
384.17948.6b R 6 6 −2 −2 0 0 2 −2 2 −2 2 2 0 0 0 −2 −2 0 0 0 0 0 0 0 0 0 0 0
384.17948.6c R 6 6 −2 −2 0 0 −2 2 −2 2 0 0 −2 −2 0 0 0 0 0 2 2 0 0 0 0 0 0 0
384.17948.6d R 6 6 −2 −2 0 0 2 −2 2 −2 −2 −2 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0
384.17948.6e R 6 −6 2 −2 0 0 −2 −2 2 2 0 0 −2 2 0 0 0 0 0 −2 2 0 0 0 0 0 0 0
384.17948.6f R 6 −6 2 −2 0 0 −2 −2 2 2 0 0 2 −2 0 0 0 0 0 2 −2 0 0 0 0 0 0 0
384.17948.6g R 6 −6 2 −2 0 0 2 2 −2 −2 −2 2 0 0 0 2 −2 0 0 0 0 0 0 0 0 0 0 0
384.17948.6h R 6 −6 2 −2 0 0 2 2 −2 −2 2 −2 0 0 0 −2 2 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed