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Properties

Label 12T82
12T82 1 5 1->5 7 1->7 12 1->12 2 8 2->8 10 2->10 3 9 3->9 11 3->11 4 4->2 4->5 4->8 4->8 5->3 5->9 5->9 6 6->12 8->9 8->10 8->12 9->1 9->11 10->4 11->5 12->4
Degree $12$
Order $144$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_6^2:C_4$

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Copy content magma:G := TransitiveGroup(12, 82);
 

Group invariants

Abstract group:  $C_6^2:C_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $144=2^{4} \cdot 3^{2}$
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 $t$:  $82$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[(1/4.2^{3})^{2}]F_{36}(6)$
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:  $(1,12)(4,5)(8,9)$, $(1,5,9)(4,8,12)$, $(2,10)(3,11)(4,8)(5,9)$, $(1,7)(2,8,10,4)(3,9,11,5)(6,12)$
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, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$36$:  $C_3^2:C_4$
$72$:  12T40

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $D_{4}$

Degree 6: $C_3^2:C_4$

Low degree siblings

12T82 x 7, 24T272 x 4, 24T273 x 4, 24T274 x 4, 36T126 x 2, 36T141 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 $2^{3},1^{6}$ $2$ $2$ $3$ $( 1,12)( 4, 5)( 8, 9)$
2C $2^{4},1^{4}$ $9$ $2$ $4$ $( 1, 5)( 4,12)( 6,10)( 7,11)$
2D $2^{6}$ $9$ $2$ $6$ $( 1, 8)( 2, 3)( 4, 5)( 6,11)( 7,10)( 9,12)$
2E $2^{5},1^{2}$ $18$ $2$ $5$ $( 1, 4)( 5,12)( 6,10)( 7,11)( 8, 9)$
3A $3^{2},1^{6}$ $4$ $3$ $4$ $( 1, 5, 9)( 4, 8,12)$
3B $3^{4}$ $4$ $3$ $8$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
4A1 $4^{2},2^{2}$ $18$ $4$ $8$ $( 1, 6, 5,10)( 2, 9)( 3, 8)( 4,11,12, 7)$
4A-1 $4^{2},2^{2}$ $18$ $4$ $8$ $( 1,10, 5, 6)( 2, 9)( 3, 8)( 4, 7,12,11)$
4B1 $4^{3}$ $18$ $4$ $9$ $( 1, 6, 8,11)( 2, 4, 3, 5)( 7, 9,10,12)$
4B-1 $4^{3}$ $18$ $4$ $9$ $( 1,11, 8, 6)( 2, 5, 3, 4)( 7,12,10, 9)$
6A $6,1^{6}$ $4$ $6$ $5$ $( 1, 8, 5,12, 9, 4)$
6B $6^{2}$ $4$ $6$ $10$ $( 1, 4, 9,12, 5, 8)( 2, 7,10, 3, 6,11)$
6C $6,3^{2}$ $4$ $6$ $9$ $( 1, 4, 9,12, 5, 8)( 2,10, 6)( 3,11, 7)$
6D $3^{2},2^{3}$ $4$ $6$ $7$ $( 1,12)( 2, 6,10)( 3, 7,11)( 4, 5)( 8, 9)$
6E $6,2^{3}$ $4$ $6$ $8$ $( 1,12)( 2, 7,10, 3, 6,11)( 4, 5)( 8, 9)$
6F $6,3^{2}$ $4$ $6$ $9$ $( 1, 8, 5,12, 9, 4)( 2,10, 6)( 3,11, 7)$

Malle's constant $a(G)$:     $1/3$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 3A 3B 4A1 4A-1 4B1 4B-1 6A 6B 6C 6D 6E 6F
Size 1 1 2 9 9 18 4 4 18 18 18 18 4 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 3A 3B 2C 2C 2D 2D 3A 3B 3B 3A 3A 3B
3 P 1A 2A 2B 2C 2D 2E 1A 1A 4A-1 4A1 4B-1 4B1 2B 2A 2B 2B 2A 2B
Type
144.136.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
144.136.1b R 1 1 −1 1 1 −1 1 1 −1 1 1 −1 −1 −1 1 −1 −1 1
144.136.1c R 1 1 −1 1 1 −1 1 1 1 −1 −1 1 −1 −1 1 −1 −1 1
144.136.1d R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1
144.136.1e1 C 1 1 −1 −1 −1 1 1 1 −i −i i i −1 −1 1 −1 −1 1
144.136.1e2 C 1 1 −1 −1 −1 1 1 1 i i −i −i −1 −1 1 −1 −1 1
144.136.1f1 C 1 1 1 −1 −1 −1 1 1 −i i −i i 1 1 1 1 1 1
144.136.1f2 C 1 1 1 −1 −1 −1 1 1 i −i i −i 1 1 1 1 1 1
144.136.2a R 2 −2 0 −2 2 0 2 2 0 0 0 0 0 0 −2 0 0 −2
144.136.2b R 2 −2 0 2 −2 0 2 2 0 0 0 0 0 0 −2 0 0 −2
144.136.4a R 4 4 4 0 0 0 −2 1 0 0 0 0 1 −2 −2 −2 1 1
144.136.4b R 4 4 4 0 0 0 1 −2 0 0 0 0 −2 1 1 1 −2 −2
144.136.4c R 4 4 −4 0 0 0 −2 1 0 0 0 0 −1 2 −2 2 −1 1
144.136.4d R 4 4 −4 0 0 0 1 −2 0 0 0 0 2 −1 1 −1 2 −2
144.136.4e R 4 −4 0 0 0 0 −2 1 0 0 0 0 −3 0 2 0 3 −1
144.136.4f R 4 −4 0 0 0 0 −2 1 0 0 0 0 3 0 2 0 −3 −1
144.136.4g R 4 −4 0 0 0 0 1 −2 0 0 0 0 0 −3 −1 3 0 2
144.136.4h R 4 −4 0 0 0 0 1 −2 0 0 0 0 0 3 −1 −3 0 2

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed