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Properties

Label 18T190
18T190 1 4 1->4 11 1->11 2 5 2->5 12 2->12 3 6 3->6 10 3->10 14 4->14 15 5->15 13 6->13 7 17 7->17 8 18 8->18 9 16 9->16 10->2 10->12 11->10 12->3 12->11 13->8 13->18 14->7 14->16 15->9 15->17 16->5 16->15 17->4 17->13 18->6 18->14
Degree $18$
Order $648$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^4:(C_2\times C_4)$

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Copy content magma:G := TransitiveGroup(18, 190);
 

Group invariants

Abstract group:  $C_3^4:(C_2\times C_4)$
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 $n$:  $18$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $190$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
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:  $(1,11)(2,12,3,10)(4,14,7,17)(5,15,9,16)(6,13,8,18)$, $(1,4)(2,5)(3,6)(10,12,11)(13,18,14,16,15,17)$
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, $C_2^2$
$8$:  $C_4\times C_2$
$36$:  $C_3^2:C_4$ x 2
$72$:  12T40 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $C_3^2:C_4$ x 2

Degree 9: None

Low degree siblings

12T171 x 8, 18T190 x 7, 18T192 x 8, 24T1523 x 8, 36T1082 x 8, 36T1083 x 16, 36T1087 x 8, 36T1198 x 16, 36T1233 x 8

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{18}$ $1$ $1$ $0$ $()$
2A $2^{6},1^{6}$ $9$ $2$ $6$ $( 1, 3)( 4, 6)( 7, 9)(11,12)(14,15)(17,18)$
2B $2^{6},1^{6}$ $9$ $2$ $6$ $( 4, 7)( 5, 8)( 6, 9)(10,13)(11,14)(12,15)$
2C $2^{8},1^{2}$ $81$ $2$ $8$ $( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,12)(13,18)(14,17)(15,16)$
3A $3^{3},1^{9}$ $4$ $3$ $6$ $(10,16,13)(11,17,14)(12,18,15)$
3B $3^{6}$ $4$ $3$ $12$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,18,17)$
3C $3^{6}$ $4$ $3$ $12$ $( 1, 7, 4)( 2, 8, 5)( 3, 9, 6)(10,13,16)(11,14,17)(12,15,18)$
3D $3^{3},1^{9}$ $4$ $3$ $6$ $(1,3,2)(4,6,5)(7,9,8)$
3E $3^{6}$ $8$ $3$ $12$ $( 1, 6, 8)( 2, 4, 9)( 3, 5, 7)(10,16,13)(11,17,14)(12,18,15)$
3F $3^{6}$ $8$ $3$ $12$ $( 1, 9, 5)( 2, 7, 6)( 3, 8, 4)(10,16,13)(11,17,14)(12,18,15)$
3G $3^{6}$ $8$ $3$ $12$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,16,13)(11,17,14)(12,18,15)$
3H $3^{6}$ $8$ $3$ $12$ $( 1, 9, 5)( 2, 7, 6)( 3, 8, 4)(10,12,11)(13,15,14)(16,18,17)$
3I $3^{6}$ $8$ $3$ $12$ $( 1, 5, 9)( 2, 6, 7)( 3, 4, 8)(10,18,14)(11,16,15)(12,17,13)$
3J $3^{3},1^{9}$ $8$ $3$ $6$ $(10,17,15)(11,18,13)(12,16,14)$
3K $3^{6}$ $8$ $3$ $12$ $( 1, 8, 6)( 2, 9, 4)( 3, 7, 5)(10,18,14)(11,16,15)(12,17,13)$
3L $3^{6}$ $8$ $3$ $12$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,18,14)(11,16,15)(12,17,13)$
4A1 $4^{4},2$ $81$ $4$ $13$ $( 1,11)( 2,10, 3,12)( 4,14, 7,17)( 5,13, 9,18)( 6,15, 8,16)$
4A-1 $4^{4},2$ $81$ $4$ $13$ $( 1,11)( 2,12, 3,10)( 4,17, 7,14)( 5,18, 9,13)( 6,16, 8,15)$
4B1 $4^{4},2$ $81$ $4$ $13$ $( 1,18, 6,11)( 2,16, 5,10)( 3,17, 4,12)( 7,15, 9,14)( 8,13)$
4B-1 $4^{4},2$ $81$ $4$ $13$ $( 1,11, 6,18)( 2,10, 5,16)( 3,12, 4,17)( 7,14, 9,15)( 8,13)$
6A $6,3,2^{3},1^{3}$ $36$ $6$ $10$ $( 1, 3)( 4, 6)( 7, 9)(10,13,16)(11,15,17,12,14,18)$
6B $6^{2},3^{2}$ $36$ $6$ $14$ $( 1, 3, 2)( 4, 9, 5, 7, 6, 8)(10,14,12,13,11,15)(16,17,18)$
6C $6^{2},3^{2}$ $36$ $6$ $14$ $( 1, 4, 7)( 2, 6, 8, 3, 5, 9)(10,17,13,11,16,14)(12,18,15)$
6D $6,3,2^{3},1^{3}$ $36$ $6$ $10$ $( 1, 5, 3, 4, 2, 6)( 7, 8, 9)(13,16)(14,17)(15,18)$

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 4A1 4A-1 4B1 4B-1 6A 6B 6C 6D
Size 1 9 9 81 4 4 4 4 8 8 8 8 8 8 8 8 81 81 81 81 36 36 36 36
2 P 1A 1A 1A 1A 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 2C 2C 2C 2C 3A 3B 3C 3D
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 4A-1 4A1 4B-1 4B1 2A 2B 2A 2B
Type
648.717.1a 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
648.717.1b 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
648.717.1c 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
648.717.1d 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
648.717.1e1 C 1 −1 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 −i i i −i −1 1 1 −1
648.717.1e2 C 1 −1 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 i −i −i i −1 1 1 −1
648.717.1f1 C 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 −i i −i i 1 −1 −1 1
648.717.1f2 C 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 i −i i −i 1 −1 −1 1
648.717.4a R 4 0 4 0 4 −2 4 1 −2 −2 −2 1 1 −2 1 1 0 0 0 0 0 1 −2 0
648.717.4b R 4 0 4 0 4 1 4 −2 1 1 1 −2 −2 1 −2 −2 0 0 0 0 0 −2 1 0
648.717.4c R 4 4 0 0 −2 4 1 4 −2 1 −2 1 −2 1 −2 1 0 0 0 0 −2 0 0 1
648.717.4d R 4 4 0 0 1 4 −2 4 1 −2 1 −2 1 −2 1 −2 0 0 0 0 1 0 0 −2
648.717.4e R 4 −4 0 0 −2 4 1 4 −2 1 −2 1 −2 1 −2 1 0 0 0 0 2 0 0 −1
648.717.4f R 4 −4 0 0 1 4 −2 4 1 −2 1 −2 1 −2 1 −2 0 0 0 0 −1 0 0 2
648.717.4g R 4 0 −4 0 4 −2 4 1 −2 −2 −2 1 1 −2 1 1 0 0 0 0 0 −1 2 0
648.717.4h R 4 0 −4 0 4 1 4 −2 1 1 1 −2 −2 1 −2 −2 0 0 0 0 0 2 −1 0
648.717.8a R 8 0 0 0 −4 −4 2 2 2 −1 2 −4 −1 −1 −1 5 0 0 0 0 0 0 0 0
648.717.8b R 8 0 0 0 −4 −4 2 2 2 −1 2 5 −1 −1 −1 −4 0 0 0 0 0 0 0 0
648.717.8c R 8 0 0 0 −4 2 2 −4 −1 −4 −1 −1 2 5 2 −1 0 0 0 0 0 0 0 0
648.717.8d R 8 0 0 0 −4 2 2 −4 −1 5 −1 −1 2 −4 2 −1 0 0 0 0 0 0 0 0
648.717.8e R 8 0 0 0 2 −4 −4 2 −1 2 −1 −1 −4 2 5 −1 0 0 0 0 0 0 0 0
648.717.8f R 8 0 0 0 2 −4 −4 2 −1 2 −1 −1 5 2 −4 −1 0 0 0 0 0 0 0 0
648.717.8g R 8 0 0 0 2 2 −4 −4 −4 −1 5 2 −1 −1 −1 2 0 0 0 0 0 0 0 0
648.717.8h R 8 0 0 0 2 2 −4 −4 5 −1 −4 2 −1 −1 −1 2 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed