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Properties

Label 12T198
12T198 1 5 1->5 6 1->6 12 1->12 2 3 2->3 8 2->8 10 2->10 9 3->9 11 3->11 4 4->2 4->8 4->8 5->3 5->9 5->9 6->12 7 7->1 8->10 8->12 9->1 9->11 10->4 11->5 12->4 12->7
Degree $12$
Order $1152$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times A_4^2:C_4$

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Copy content magma:G := TransitiveGroup(12, 198);
 

Group invariants

Abstract group:  $C_2\times A_4^2:C_4$
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 $n$:  $12$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $198$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $1/2[2^{6}]F_{36}$
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:  $(1,12)(2,3)$, $(1,5,9)(4,8,12)$, $(1,6,12,7)(2,8,10,4)(3,9,11,5)$, $(2,10)(3,11)(4,8)(5,9)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$36$:  $C_3^2:C_4$
$72$:  12T40
$576$:  $A_4^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $C_3^2:C_4$

Low degree siblings

12T199, 16T1287, 16T1290, 18T265 x 2, 24T2781, 24T2798, 24T2799, 24T2800, 24T2801, 24T2802, 32T96689, 32T96706 x 2, 36T1664, 36T1665, 36T1666, 36T1667, 36T1707 x 2, 36T1708 x 2, 36T1930, 36T1932

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 $2^{2},1^{8}$ $6$ $2$ $2$ $( 2, 3)(10,11)$
2C $2^{4},1^{4}$ $6$ $2$ $4$ $( 1,12)( 2, 3)( 6, 7)(10,11)$
2D $2^{2},1^{8}$ $9$ $2$ $2$ $( 8, 9)(10,11)$
2E $2^{4},1^{4}$ $9$ $2$ $4$ $( 1,12)( 2, 3)( 8, 9)(10,11)$
2F $2^{6}$ $36$ $2$ $6$ $( 1,12)( 2, 3)( 4, 8)( 5, 9)( 6,10)( 7,11)$
2G $2^{4},1^{4}$ $36$ $2$ $4$ $( 1, 9)( 2,10)( 3,11)( 8,12)$
3A $3^{2},1^{6}$ $16$ $3$ $4$ $( 2, 6,11)( 3, 7,10)$
3B $3^{4}$ $64$ $3$ $8$ $( 1, 9, 5)( 2, 6,11)( 3, 7,10)( 4,12, 8)$
4A $4^{2},1^{4}$ $36$ $4$ $6$ $( 1, 8,12, 9)( 2,11, 3,10)$
4B $4^{2},2^{2}$ $36$ $4$ $8$ $( 1,12)( 2, 3)( 4, 9, 5, 8)( 6,10, 7,11)$
4C $4,2^{3},1^{2}$ $72$ $4$ $6$ $( 1, 5,12, 4)( 2,10)( 3,11)( 6, 7)$
4D $4,2^{3},1^{2}$ $72$ $4$ $6$ $( 1,12)( 2, 7)( 3, 6)( 4, 9, 5, 8)$
4E1 $4^{3}$ $72$ $4$ $9$ $( 1, 3,12, 2)( 4, 7, 8,11)( 5, 6, 9,10)$
4E-1 $4^{3}$ $72$ $4$ $9$ $( 1, 2,12, 3)( 4,11, 8, 7)( 5,10, 9, 6)$
4F1 $4^{3}$ $72$ $4$ $9$ $( 1, 2, 9,11)( 3, 8,10,12)( 4, 6, 5, 7)$
4F-1 $4^{3}$ $72$ $4$ $9$ $( 1,11, 9, 2)( 3,12,10, 8)( 4, 7, 5, 6)$
6A $6,2^{3}$ $16$ $6$ $8$ $( 1,12)( 2,10, 6, 3,11, 7)( 4, 5)( 8, 9)$
6B $3^{2},2^{2},1^{2}$ $48$ $6$ $6$ $( 1, 4, 8)( 2, 3)( 5, 9,12)(10,11)$
6C $6,2,1^{4}$ $48$ $6$ $6$ $( 1,12)( 2, 7,11, 3, 6,10)$
6D $6^{2}$ $64$ $6$ $10$ $( 1, 4, 9,12, 5, 8)( 2,10, 6, 3,11, 7)$
8A1 $8,2^{2}$ $72$ $8$ $9$ $( 1, 2, 8,11,12, 3, 9,10)( 4, 7)( 5, 6)$
8A-1 $8,2^{2}$ $72$ $8$ $9$ $( 1,10, 9, 3,12,11, 8, 2)( 4, 7)( 5, 6)$
8B1 $8,2^{2}$ $72$ $8$ $9$ $( 1, 3, 8,10,12, 2, 9,11)( 4, 6)( 5, 7)$
8B-1 $8,2^{2}$ $72$ $8$ $9$ $( 1,11, 9, 2,12,10, 8, 3)( 4, 6)( 5, 7)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E1 4E-1 4F1 4F-1 6A 6B 6C 6D 8A1 8A-1 8B1 8B-1
Size 1 1 6 6 9 9 36 36 16 64 36 36 72 72 72 72 72 72 16 48 48 64 72 72 72 72
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A 3B 2E 2E 2B 2B 2F 2F 2F 2F 3A 3A 3A 3B 4A 4A 4A 4A
3 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 1A 4A 4B 4C 4D 4E-1 4E1 4F-1 4F1 2A 2B 2C 2A 8A-1 8A1 8B-1 8B1
Type
1152.157850.1a 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
1152.157850.1b 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
1152.157850.1c 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
1152.157850.1d 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
1152.157850.1e1 C 1 −1 1 −1 −1 1 −1 1 1 1 −1 1 −i 1 i i −1 −i −1 1 −1 −1 i −i −i i
1152.157850.1e2 C 1 −1 1 −1 −1 1 −1 1 1 1 −1 1 i 1 −i −i −1 i −1 1 −1 −1 −i i i −i
1152.157850.1f1 C 1 1 1 1 1 1 −1 −1 1 1 −1 −1 −i −1 −i i −1 i 1 1 1 1 −i i −i i
1152.157850.1f2 C 1 1 1 1 1 1 −1 −1 1 1 −1 −1 i −1 i −i −1 −i 1 1 1 1 i −i i −i
1152.157850.4a R 4 4 4 4 4 4 0 0 −2 1 0 0 0 0 0 0 0 0 −2 −2 −2 1 0 0 0 0
1152.157850.4b R 4 4 4 4 4 4 0 0 1 −2 0 0 0 0 0 0 0 0 1 1 1 −2 0 0 0 0
1152.157850.4c R 4 −4 4 −4 −4 4 0 0 −2 1 0 0 0 0 0 0 0 0 2 −2 2 −1 0 0 0 0
1152.157850.4d R 4 −4 4 −4 −4 4 0 0 1 −2 0 0 0 0 0 0 0 0 −1 1 −1 2 0 0 0 0
1152.157850.6a R 6 6 2 2 −2 −2 2 2 3 0 −2 −2 0 0 0 0 0 0 3 −1 −1 0 0 0 0 0
1152.157850.6b R 6 6 2 2 −2 −2 −2 −2 3 0 2 2 0 0 0 0 0 0 3 −1 −1 0 0 0 0 0
1152.157850.6c R 6 −6 2 −2 2 −2 −2 2 3 0 2 −2 0 0 0 0 0 0 −3 −1 1 0 0 0 0 0
1152.157850.6d R 6 −6 2 −2 2 −2 2 −2 3 0 −2 2 0 0 0 0 0 0 −3 −1 1 0 0 0 0 0
1152.157850.9a R 9 9 −3 −3 1 1 1 1 0 0 1 1 1 −1 1 1 −1 1 0 0 0 0 −1 −1 −1 −1
1152.157850.9b R 9 9 −3 −3 1 1 1 1 0 0 1 1 −1 −1 −1 −1 −1 −1 0 0 0 0 1 1 1 1
1152.157850.9c R 9 −9 −3 3 −1 1 1 −1 0 0 1 −1 −1 1 1 −1 −1 1 0 0 0 0 −1 −1 1 1
1152.157850.9d R 9 −9 −3 3 −1 1 1 −1 0 0 1 −1 1 1 −1 1 −1 −1 0 0 0 0 1 1 −1 −1
1152.157850.9e1 C 9 9 −3 −3 1 1 −1 −1 0 0 −1 −1 −i 1 −i i 1 i 0 0 0 0 i −i i −i
1152.157850.9e2 C 9 9 −3 −3 1 1 −1 −1 0 0 −1 −1 i 1 i −i 1 −i 0 0 0 0 −i i −i i
1152.157850.9f1 C 9 −9 −3 3 −1 1 −1 1 0 0 −1 1 −i −1 i i 1 −i 0 0 0 0 −i i i −i
1152.157850.9f2 C 9 −9 −3 3 −1 1 −1 1 0 0 −1 1 i −1 −i −i 1 i 0 0 0 0 i −i −i i
1152.157850.12a R 12 12 4 4 −4 −4 0 0 −3 0 0 0 0 0 0 0 0 0 −3 1 1 0 0 0 0 0
1152.157850.12b R 12 −12 4 −4 4 −4 0 0 −3 0 0 0 0 0 0 0 0 0 3 1 −1 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed