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Properties

Label 12T71
12T71 1 7 1->7 10 1->10 2 5 2->5 6 2->6 8 2->8 3 3->7 9 3->9 12 3->12 4 4->7 4->8 4->10 11 5->11 6->9 6->10 6->12 7->11 8->11 8->12 10->2 11->3 12->4
Degree $12$
Order $108$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3:S_3^2$

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Copy content magma:G := TransitiveGroup(12, 71);
 

Group invariants

Abstract group:  $C_3:S_3^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $108=2^{2} \cdot 3^{3}$
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 $t$:  $71$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[3^{3}]E(4)$
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $3$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,10)(2,5)(3,12)(4,7)(6,9)(8,11)$, $(2,6,10)(3,7,11)(4,8,12)$, $(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 3
$12$:  $D_{6}$ x 3
$36$:  $S_3^2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: None

Low degree siblings

18T53 x 3, 27T35, 36T88 x 3, 36T93

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 $2^{6}$ $9$ $2$ $6$ $( 1,10)( 2, 5)( 3, 8)( 4,11)( 6, 9)( 7,12)$
2B $2^{6}$ $9$ $2$ $6$ $( 1, 8)( 2, 7)( 3,10)( 4, 9)( 5,12)( 6,11)$
2C $2^{6}$ $9$ $2$ $6$ $( 1, 3)( 2, 8)( 4,10)( 5, 7)( 6,12)( 9,11)$
3A $3^{4}$ $2$ $3$ $8$ $( 1, 5, 9)( 2, 6,10)( 3,11, 7)( 4,12, 8)$
3B $3^{4}$ $2$ $3$ $8$ $( 1, 5, 9)( 2,10, 6)( 3,11, 7)( 4, 8,12)$
3C $3^{4}$ $2$ $3$ $8$ $( 1, 9, 5)( 2, 6,10)( 3,11, 7)( 4, 8,12)$
3D $3^{2},1^{6}$ $4$ $3$ $4$ $( 2, 6,10)( 4,12, 8)$
3E $3^{2},1^{6}$ $4$ $3$ $4$ $( 3,11, 7)( 4, 8,12)$
3F $3^{2},1^{6}$ $4$ $3$ $4$ $( 2, 6,10)( 3,11, 7)$
3G1 $3^{3},1^{3}$ $4$ $3$ $6$ $( 1, 5, 9)( 3, 7,11)( 4, 8,12)$
3G-1 $3^{3},1^{3}$ $4$ $3$ $6$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)$
6A $6^{2}$ $18$ $6$ $10$ $( 1, 6, 5,10, 9, 2)( 3,12,11, 8, 7, 4)$
6B $6^{2}$ $18$ $6$ $10$ $( 1, 4, 5, 8, 9,12)( 2,11,10, 7, 6, 3)$
6C $6^{2}$ $18$ $6$ $10$ $( 1, 7, 9, 3, 5,11)( 2, 4, 6, 8,10,12)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G1 3G-1 6A 6B 6C
Size 1 9 9 9 2 2 2 4 4 4 4 4 18 18 18
2 P 1A 1A 1A 1A 3A 3B 3C 3D 3E 3F 3G-1 3G1 3A 3B 3C
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 1A 1A 1A 2A 2B 2C
Type
108.40.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
108.40.1b R 1 −1 −1 1 1 1 1 1 1 1 1 1 −1 −1 1
108.40.1c R 1 −1 1 −1 1 1 1 1 1 1 1 1 −1 1 −1
108.40.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 −1 −1
108.40.2a R 2 0 0 2 2 2 −1 −1 2 −1 −1 −1 0 0 −1
108.40.2b R 2 0 2 0 2 −1 2 2 −1 −1 −1 −1 0 −1 0
108.40.2c R 2 2 0 0 −1 2 2 −1 −1 2 −1 −1 −1 0 0
108.40.2d R 2 −2 0 0 −1 2 2 −1 −1 2 −1 −1 1 0 0
108.40.2e R 2 0 −2 0 2 −1 2 2 −1 −1 −1 −1 0 1 0
108.40.2f R 2 0 0 −2 2 2 −1 −1 2 −1 −1 −1 0 0 1
108.40.4a R 4 0 0 0 −2 −2 4 −2 1 −2 1 1 0 0 0
108.40.4b R 4 0 0 0 −2 4 −2 1 −2 −2 1 1 0 0 0
108.40.4c R 4 0 0 0 4 −2 −2 −2 −2 1 1 1 0 0 0
108.40.4d1 C 4 0 0 0 −2 −2 −2 1 1 1 −2−3ζ3 1+3ζ3 0 0 0
108.40.4d2 C 4 0 0 0 −2 −2 −2 1 1 1 1+3ζ3 −2−3ζ3 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed