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Properties

Label 12T237
12T237 1 3 1->3 12 1->12 2 2->3 4 2->4 6 2->6 5 3->5 7 3->7 4->6 8 4->8 5->7 9 5->9 10 6->10 6->12 7->1 11 7->11 8->10 9->11 10->2 10->9 11->3 11->8 12->2
Degree $12$
Order $2304$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_4^2:C_4$

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Copy content magma:G := TransitiveGroup(12, 237);
 

Group invariants

Abstract group:  $S_4^2:C_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $2304=2^{8} \cdot 3^{2}$
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 $n$:  $12$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $237$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[2^{5}]F_{36}:2_{2}{S_{3}^{2},i}$
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:  $(1,12)(2,3)$, $(1,3,5,7)(2,4,6,12)(8,10,9,11)$, $(2,6,10)(3,7,11)$, $(4,8)(5,9)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$72$:  $C_3^2:D_4$
$144$:  12T79
$1152$:  $S_4\wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $C_3^2:D_4$

Low degree siblings

12T237, 12T238 x 2, 16T1496 x 2, 16T1497 x 2, 24T5093 x 2, 24T5117 x 2, 24T5118 x 2, 24T5119, 24T5120 x 2, 24T5121 x 2, 24T5122 x 2, 24T5123 x 2, 24T5124, 24T5125 x 2, 24T5126 x 2, 24T5127 x 2, 24T5128 x 2, 32T205436 x 2, 32T205437 x 2, 32T205438, 32T205439, 36T3213 x 2, 36T3215 x 2, 36T3216 x 2, 36T3218, 36T3224, 36T3449 x 2, 36T3450 x 2

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 $2^{2},1^{8}$ $6$ $2$ $2$ $( 6, 7)(10,11)$
2C $2^{4},1^{4}$ $6$ $2$ $4$ $( 2, 3)( 6, 7)( 8, 9)(10,11)$
2D $2^{4},1^{4}$ $9$ $2$ $4$ $( 1,12)( 4, 5)( 6, 7)(10,11)$
2E $2^{2},1^{8}$ $9$ $2$ $2$ $( 8, 9)(10,11)$
2F $2^{6}$ $12$ $2$ $6$ $( 1,12)( 2, 3)( 4, 5)( 6,11)( 7,10)( 8, 9)$
2G $2^{2},1^{8}$ $12$ $2$ $2$ $( 2,10)( 3,11)$
2H $2^{4},1^{4}$ $36$ $2$ $4$ $( 4, 5)( 6,10)( 7,11)( 8, 9)$
2I $2^{4},1^{4}$ $36$ $2$ $4$ $( 4, 8)( 5, 9)( 6,10)( 7,11)$
2J $2^{4},1^{4}$ $36$ $2$ $4$ $( 2, 3)( 6,10)( 7,11)( 8, 9)$
2K $2^{6}$ $36$ $2$ $6$ $( 1, 9)( 2,10)( 3,11)( 4, 5)( 6, 7)( 8,12)$
3A $3^{2},1^{6}$ $16$ $3$ $4$ $( 1, 8, 5)( 4,12, 9)$
3B $3^{4}$ $64$ $3$ $8$ $( 1, 4, 8)( 2,10, 6)( 3,11, 7)( 5, 9,12)$
4A $4,2,1^{6}$ $12$ $4$ $4$ $( 2, 3)( 6,11, 7,10)$
4B $4,2^{3},1^{2}$ $12$ $4$ $6$ $( 1,12)( 2,11, 3,10)( 4, 5)( 8, 9)$
4C1 $4^{3}$ $24$ $4$ $9$ $( 1,11,12,10)( 2, 8, 3, 9)( 4, 7, 5, 6)$
4C-1 $4^{3}$ $24$ $4$ $9$ $( 1,10,12,11)( 2, 9, 3, 8)( 4, 6, 5, 7)$
4D $4,2,1^{6}$ $36$ $4$ $4$ $( 6,10, 7,11)( 8, 9)$
4E $4,2^{3},1^{2}$ $36$ $4$ $6$ $( 2, 3)( 4, 5)( 6,10, 7,11)( 8, 9)$
4F $4^{2},2^{2}$ $36$ $4$ $8$ $( 1, 4,12, 5)( 2, 3)( 6,10, 7,11)( 8, 9)$
4G $4^{2},1^{4}$ $36$ $4$ $6$ $( 2,11, 3,10)( 4, 8, 5, 9)$
4H $4,2^{3},1^{2}$ $72$ $4$ $6$ $( 1,12)( 2, 6)( 3, 7)( 4, 8, 5, 9)$
4I $4,2^{3},1^{2}$ $72$ $4$ $6$ $( 1, 5)( 2,10, 3,11)( 4,12)( 8, 9)$
4J1 $4,2^{4}$ $72$ $4$ $7$ $( 1, 3)( 2,12)( 4, 7)( 5, 6)( 8,11, 9,10)$
4J-1 $4,2^{4}$ $72$ $4$ $7$ $( 1, 3)( 2,12)( 4, 7)( 5, 6)( 8,10, 9,11)$
4K1 $4^{3}$ $144$ $4$ $9$ $( 1, 2, 9,10)( 3, 8,11,12)( 4, 7, 5, 6)$
4K-1 $4^{3}$ $144$ $4$ $9$ $( 1,10, 9, 2)( 3,12,11, 8)( 4, 6, 5, 7)$
6A $6,2^{3}$ $16$ $6$ $8$ $( 1, 4, 8,12, 5, 9)( 2, 3)( 6, 7)(10,11)$
6B $6,2,1^{4}$ $48$ $6$ $6$ $( 2, 6,10, 3, 7,11)( 8, 9)$
6C $3^{2},2^{2},1^{2}$ $48$ $6$ $6$ $( 1, 9, 4)( 5,12, 8)( 6, 7)(10,11)$
6D $6^{2}$ $64$ $6$ $10$ $( 1, 9, 4,12, 8, 5)( 2, 7,10, 3, 6,11)$
6E $6,2^{3}$ $96$ $6$ $8$ $( 1, 9, 4,12, 8, 5)( 2, 3)( 6,11)( 7,10)$
6F $3^{2},2^{2},1^{2}$ $96$ $6$ $6$ $( 1, 9, 4)( 2,10)( 3,11)( 5,12, 8)$
8A1 $8,2^{2}$ $144$ $8$ $9$ $( 1, 7)( 2, 9,11, 4, 3, 8,10, 5)( 6,12)$
8A-1 $8,2^{2}$ $144$ $8$ $9$ $( 1, 7)( 2, 5,10, 8, 3, 4,11, 9)( 6,12)$
12A $4,3^{2},2$ $96$ $12$ $8$ $( 1, 4, 9)( 2, 3)( 5, 8,12)( 6,10, 7,11)$
12B $6,4,1^{2}$ $96$ $12$ $8$ $( 1, 4, 8,12, 5, 9)( 2,10, 3,11)$
12C1 $12$ $192$ $12$ $11$ $( 1, 6, 9,11, 4, 2,12, 7, 8,10, 5, 3)$
12C-1 $12$ $192$ $12$ $11$ $( 1, 3, 5,10, 8, 7,12, 2, 4,11, 9, 6)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

40 x 40 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed