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Properties

Label 12T169
12T169 1 4 1->4 11 1->11 2 5 2->5 10 2->10 3 6 3->6 9 3->9 7 4->7 8 4->8 4->8 5->7 5->8 6->9 7->10 8->11 12 8->12 9->12 10->1 11->2 12->3 12->4
Degree $12$
Order $648$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^3:D_{12}$

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Copy content magma:G := TransitiveGroup(12, 169);
 

Group invariants

Abstract group:  $C_3^3:D_{12}$
Copy content magma:IdentifyGroup(G);
 
Order:  $648=2^{3} \cdot 3^{4}$
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 $t$:  $169$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $1/2[3^{4}:2]cD(4)$
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,11)(2,10)(3,9)(4,8)(5,7)$, $(4,8,12)$, $(1,4,7,10)(2,5,8,11)(3,6,9,12)$
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 2
$8$:  $D_{4}$
$12$:  $D_{6}$ x 2
$24$:  $D_{12}$, $(C_6\times C_2):C_2$
$36$:  $S_3^2$
$72$:  $C_3^2:D_4$, 12T38
$216$:  12T116, 12T118

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $D_{4}$

Degree 6: None

Low degree siblings

12T169, 18T210 x 2, 24T1521 x 2, 24T1533, 36T1105 x 2, 36T1106 x 2, 36T1107 x 2, 36T1153, 36T1154, 36T1160, 36T1167, 36T1173, 36T1190 x 2, 36T1195 x 2, 36T1225 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 $2^{6}$ $9$ $2$ $6$ $( 1, 3)( 2,12)( 4, 6)( 5, 7)( 8,10)( 9,11)$
2B $2^{6}$ $18$ $2$ $6$ $( 1, 6)( 2, 5)( 3, 4)( 7,12)( 8,11)( 9,10)$
2C $2^{5},1^{2}$ $54$ $2$ $5$ $( 1, 5)( 2, 4)( 6,12)( 7,11)( 8,10)$
3A $3^{4}$ $2$ $3$ $8$ $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$
3B $3^{4}$ $2$ $3$ $8$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
3C $3^{2},1^{6}$ $4$ $3$ $4$ $( 2, 6,10)( 4,12, 8)$
3D $3^{4}$ $4$ $3$ $8$ $( 1, 9, 5)( 2,10, 6)( 3, 7,11)( 4, 8,12)$
3E $3^{2},1^{6}$ $4$ $3$ $4$ $( 1, 5, 9)( 3, 7,11)$
3F1 $3^{2},1^{6}$ $4$ $3$ $4$ $( 2,10, 6)( 3, 7,11)$
3F-1 $3^{2},1^{6}$ $4$ $3$ $4$ $( 2, 6,10)( 3,11, 7)$
3G $3,1^{9}$ $8$ $3$ $2$ $(1,9,5)$
3H $3^{2},1^{6}$ $8$ $3$ $4$ $( 1, 9, 5)( 2,10, 6)$
3I $3^{3},1^{3}$ $8$ $3$ $6$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)$
3J $3^{3},1^{3}$ $8$ $3$ $6$ $( 1, 5, 9)( 2,10, 6)( 3, 7,11)$
3K $3^{4}$ $8$ $3$ $8$ $( 1, 5, 9)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
3L1 $3^{3},1^{3}$ $8$ $3$ $6$ $( 1, 5, 9)( 2,10, 6)( 3,11, 7)$
3L-1 $3^{3},1^{3}$ $8$ $3$ $6$ $( 1, 9, 5)( 2,10, 6)( 3, 7,11)$
4A $4^{3}$ $54$ $4$ $9$ $( 1,12,11, 6)( 2, 9, 8, 7)( 3,10, 5, 4)$
6A $6^{2}$ $18$ $6$ $10$ $( 1,11, 5, 3, 9, 7)( 2, 4,10,12, 6, 8)$
6B $6^{2}$ $18$ $6$ $10$ $( 1, 3, 9,11, 5, 7)( 2,12,10, 8, 6, 4)$
6C1 $6^{2}$ $18$ $6$ $10$ $( 1,10, 5, 6, 9, 2)( 3, 8, 7, 4,11,12)$
6C-1 $6^{2}$ $18$ $6$ $10$ $( 1, 6, 9,10, 5, 2)( 3, 4,11, 8, 7,12)$
6D $6^{2}$ $36$ $6$ $10$ $( 1, 4, 9, 8, 5,12)( 2,11,10, 3, 6, 7)$
6E $6,2^{3}$ $36$ $6$ $8$ $( 1, 3, 5, 7, 9,11)( 2,12)( 4, 6)( 8,10)$
6F1 $6,2^{3}$ $36$ $6$ $8$ $( 1,12)( 2, 7,10,11, 6, 3)( 4, 9)( 5, 8)$
6F-1 $6,2^{3}$ $36$ $6$ $8$ $( 1,12)( 2, 3, 6,11,10, 7)( 4, 9)( 5, 8)$
6G $6,2^{2},1^{2}$ $108$ $6$ $7$ $( 1, 5)( 2, 8, 6, 4,10,12)( 7,11)$
12A1 $12$ $54$ $12$ $11$ $( 1, 2, 3,12, 9,10,11, 8, 5, 6, 7, 4)$
12A5 $12$ $54$ $12$ $11$ $( 1,10, 7,12, 5, 2,11, 4, 9, 6, 3, 8)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 3B 3C 3D 3E 3F1 3F-1 3G 3H 3I 3J 3K 3L1 3L-1 4A 6A 6B 6C1 6C-1 6D 6E 6F1 6F-1 6G 12A1 12A5
Size 1 9 18 54 2 2 4 4 4 4 4 8 8 8 8 8 8 8 54 18 18 18 18 36 36 36 36 108 54 54
2 P 1A 1A 1A 1A 3A 3B 3C 3D 3E 3F-1 3F1 3G 3H 3I 3J 3K 3L-1 3L1 2A 3A 3B 3A 3A 3D 3E 3F1 3F-1 3C 6B 6B
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 4A 2A 2A 2B 2B 2B 2A 2B 2B 2C 4A 4A
Type
648.723.1a 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 1 1
648.723.1b 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 1 1
648.723.1c 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 −1 −1
648.723.1d 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 −1 −1
648.723.2a R 2 2 0 0 2 −1 2 2 −1 2 2 −1 2 −1 −1 −1 −1 −1 2 0 2 0 −1 0 −1 0 0 0 −1 −1
648.723.2b R 2 2 2 0 −1 2 2 2 −1 −1 −1 −1 −1 −1 2 −1 2 −1 0 −1 −1 −1 2 2 −1 −1 −1 0 0 0
648.723.2c R 2 −2 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 0 −2 0 −2 0 −2 0 0 0 0 0
648.723.2d R 2 2 −2 0 −1 2 2 2 −1 −1 −1 −1 −1 −1 2 −1 2 −1 0 1 −1 1 2 −2 −1 1 1 0 0 0
648.723.2e R 2 2 0 0 2 −1 2 2 −1 2 2 −1 2 −1 −1 −1 −1 −1 −2 0 2 0 −1 0 −1 0 0 0 1 1
648.723.2f1 R 2 −2 0 0 2 −1 2 2 −1 2 2 −1 2 −1 −1 −1 −1 −1 0 0 −2 0 1 0 1 0 0 0 −ζ12−1−ζ12 ζ12−1+ζ12
648.723.2f2 R 2 −2 0 0 2 −1 2 2 −1 2 2 −1 2 −1 −1 −1 −1 −1 0 0 −2 0 1 0 1 0 0 0 ζ12−1+ζ12 −ζ12−1−ζ12
648.723.2g1 C 2 −2 0 0 −1 2 2 2 −1 −1 −1 −1 −1 −1 2 −1 2 −1 0 −1−2ζ3 1 1+2ζ3 −2 0 1 1+2ζ3 −1−2ζ3 0 0 0
648.723.2g2 C 2 −2 0 0 −1 2 2 2 −1 −1 −1 −1 −1 −1 2 −1 2 −1 0 1+2ζ3 1 −1−2ζ3 −2 0 1 −1−2ζ3 1+2ζ3 0 0 0
648.723.4a R 4 4 0 0 −2 −2 4 4 1 −2 −2 1 −2 1 −2 1 −2 1 0 0 −2 0 −2 0 1 0 0 0 0 0
648.723.4b R 4 0 0 2 4 4 1 −2 4 −2 −2 1 1 −2 1 −2 −2 1 0 0 0 0 0 0 0 0 0 −1 0 0
648.723.4c R 4 0 2 0 4 4 −2 1 4 1 1 −2 −2 1 −2 1 1 −2 0 2 0 2 0 −1 0 −1 −1 0 0 0
648.723.4d R 4 0 −2 0 4 4 −2 1 4 1 1 −2 −2 1 −2 1 1 −2 0 −2 0 −2 0 1 0 1 1 0 0 0
648.723.4e R 4 0 0 −2 4 4 1 −2 4 −2 −2 1 1 −2 1 −2 −2 1 0 0 0 0 0 0 0 0 0 1 0 0
648.723.4f R 4 −4 0 0 −2 −2 4 4 1 −2 −2 1 −2 1 −2 1 −2 1 0 0 2 0 2 0 −1 0 0 0 0 0
648.723.4g1 C 4 0 2 0 −2 4 −2 1 −2 −2−3ζ3 1+3ζ3 1 1 1+3ζ3 −2 −2−3ζ3 1 1 0 2ζ3 0 2ζ3−1 0 −1 0 −ζ3−1 −ζ3 0 0 0
648.723.4g2 C 4 0 2 0 −2 4 −2 1 −2 1+3ζ3 −2−3ζ3 1 1 −2−3ζ3 −2 1+3ζ3 1 1 0 2ζ3−1 0 2ζ3 0 −1 0 −ζ3 −ζ3−1 0 0 0
648.723.4h1 C 4 0 −2 0 −2 4 −2 1 −2 −2−3ζ3 1+3ζ3 1 1 1+3ζ3 −2 −2−3ζ3 1 1 0 −2ζ3 0 −2ζ3−1 0 1 0 ζ3−1 ζ3 0 0 0
648.723.4h2 C 4 0 −2 0 −2 4 −2 1 −2 1+3ζ3 −2−3ζ3 1 1 −2−3ζ3 −2 1+3ζ3 1 1 0 −2ζ3−1 0 −2ζ3 0 1 0 ζ3 ζ3−1 0 0 0
648.723.8a R 8 0 0 0 8 −4 −4 2 −4 2 2 2 −4 −1 2 −1 −1 2 0 0 0 0 0 0 0 0 0 0 0 0
648.723.8b R 8 0 0 0 8 −4 2 −4 −4 −4 −4 −1 2 2 −1 2 2 −1 0 0 0 0 0 0 0 0 0 0 0 0
648.723.8c R 8 0 0 0 −4 8 2 −4 −4 2 2 −1 −1 2 2 2 −4 −1 0 0 0 0 0 0 0 0 0 0 0 0
648.723.8d R 8 0 0 0 −4 −4 2 −4 2 2 2 −4 −1 −1 −1 −1 2 5 0 0 0 0 0 0 0 0 0 0 0 0
648.723.8e R 8 0 0 0 −4 −4 2 −4 2 2 2 5 −1 −1 −1 −1 2 −4 0 0 0 0 0 0 0 0 0 0 0 0
648.723.8f1 C 8 0 0 0 −4 −4 −4 2 2 −4−6ζ3 2+6ζ3 −1 2 −1−3ζ3 2 2+3ζ3 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0
648.723.8f2 C 8 0 0 0 −4 −4 −4 2 2 2+6ζ3 −4−6ζ3 −1 2 2+3ζ3 2 −1−3ζ3 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed