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Properties

Label 18T45
18T45 1 18 1->18 2 17 2->17 3 4 3->4 6 3->6 5 4->5 8 5->8 9 5->9 7 6->7 10 6->10 12 7->12 13 7->13 11 8->11 14 8->14 15 9->15 9->18 16 10->16 10->17 11->1 11->4 12->2 12->3 13->6 14->5 15->12 16->11 17->13 17->15 18->14 18->16
Degree $18$
Order $108$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{18}:C_6$

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Copy content magma:G := TransitiveGroup(18, 45);
 

Group invariants

Abstract group:  $C_{18}:C_6$
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 $n$:  $18$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $45$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(3,6,10,17,15,12)(4,5,9,18,16,11)(7,13)(8,14)$, $(1,18,14,5,8,11)(2,17,13,6,7,12)(3,4)(9,15)(10,16)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$36$:  $C_6\times S_3$
$54$:  $(C_9:C_3):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 9: $(C_9:C_3):C_2$

Low degree siblings

18T45, 36T67

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 $2^{9}$ $1$ $2$ $9$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
2B $2^{8},1^{2}$ $9$ $2$ $8$ $( 3,17)( 4,18)( 5,16)( 6,15)( 7,13)( 8,14)( 9,11)(10,12)$
2C $2^{9}$ $9$ $2$ $9$ $( 1, 4)( 2, 3)( 5,17)( 6,18)( 7,15)( 8,16)( 9,14)(10,13)(11,12)$
3A $3^{6}$ $2$ $3$ $12$ $( 1,14, 8)( 2,13, 7)( 3,15,10)( 4,16, 9)( 5,18,11)( 6,17,12)$
3B1 $3^{4},1^{6}$ $3$ $3$ $8$ $( 1,14, 8)( 2,13, 7)( 3,10,15)( 4, 9,16)$
3B-1 $3^{4},1^{6}$ $3$ $3$ $8$ $( 1, 8,14)( 2, 7,13)( 3,15,10)( 4,16, 9)$
6A $6^{3}$ $2$ $6$ $15$ $( 1, 7,14, 2, 8,13)( 3, 9,15, 4,10,16)( 5,12,18, 6,11,17)$
6B1 $6^{2},2^{3}$ $3$ $6$ $13$ $( 1, 7,14, 2, 8,13)( 3,16,10, 4,15, 9)( 5, 6)(11,12)(17,18)$
6B-1 $6^{2},2^{3}$ $3$ $6$ $13$ $( 1, 7,14, 2, 8,13)( 3, 4)( 5,17,11, 6,18,12)( 9,10)(15,16)$
6C1 $6^{2},2^{2},1^{2}$ $9$ $6$ $12$ $( 3, 6,10,17,15,12)( 4, 5, 9,18,16,11)( 7,13)( 8,14)$
6C-1 $6^{2},2^{2},1^{2}$ $9$ $6$ $12$ $( 3,12,15,17,10, 6)( 4,11,16,18, 9, 5)( 7,13)( 8,14)$
6D1 $6^{2},2^{3}$ $9$ $6$ $13$ $( 1, 9, 8, 4,14,16)( 2,10, 7, 3,13,15)( 5,17)( 6,18)(11,12)$
6D-1 $6^{2},2^{3}$ $9$ $6$ $13$ $( 1, 9,14,16, 8, 4)( 2,10,13,15, 7, 3)( 5,12)( 6,11)(17,18)$
9A $9^{2}$ $6$ $9$ $16$ $( 1,17,15,14,12,10, 8, 6, 3)( 2,18,16,13,11, 9, 7, 5, 4)$
9B1 $9^{2}$ $6$ $9$ $16$ $( 1, 6,15,14,17,10, 8,12, 3)( 2, 5,16,13,18, 9, 7,11, 4)$
9B-1 $9^{2}$ $6$ $9$ $16$ $( 1, 3,12, 8,10,17,14,15, 6)( 2, 4,11, 7, 9,18,13,16, 5)$
18A $18$ $6$ $18$ $17$ $( 1, 9,17, 7,15, 5,14, 4,12, 2,10,18, 8,16, 6,13, 3,11)$
18B1 $18$ $6$ $18$ $17$ $( 1, 9, 6, 7,15,11,14, 4,17, 2,10, 5, 8,16,12,13, 3,18)$
18B-1 $18$ $6$ $18$ $17$ $( 1, 9,12, 7,15,18,14, 4, 6, 2,10,11, 8,16,17,13, 3, 5)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 3B1 3B-1 6A 6B1 6B-1 6C1 6C-1 6D1 6D-1 9A 9B1 9B-1 18A 18B1 18B-1
Size 1 1 9 9 2 3 3 2 3 3 9 9 9 9 6 6 6 6 6 6
2 P 1A 1A 1A 1A 3A 3B-1 3B1 3A 3B1 3B-1 3B-1 3B1 3B-1 3B1 9A 9B-1 9B1 9A 9B1 9B-1
3 P 1A 2A 2B 2C 1A 1A 1A 2A 2A 2A 2B 2B 2C 2C 3A 3A 3A 6A 6A 6A
Type
108.26.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
108.26.1b R 1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1 −1 1 1 1 −1 −1 −1
108.26.1c R 1 −1 1 −1 1 1 1 −1 −1 −1 1 −1 −1 1 1 1 1 −1 −1 −1
108.26.1d R 1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1
108.26.1e1 C 1 1 1 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 1 ζ3−1 ζ3 1 ζ3 ζ3−1
108.26.1e2 C 1 1 1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 1 ζ3 ζ3−1 1 ζ3−1 ζ3
108.26.1f1 C 1 −1 −1 1 1 ζ3−1 ζ3 −1 −ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3−1 1 ζ3−1 ζ3 −1 −ζ3 −ζ3−1
108.26.1f2 C 1 −1 −1 1 1 ζ3 ζ3−1 −1 −ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3 1 ζ3 ζ3−1 −1 −ζ3−1 −ζ3
108.26.1g1 C 1 −1 1 −1 1 ζ3−1 ζ3 −1 −ζ3 −ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3−1 1 ζ3−1 ζ3 −1 −ζ3 −ζ3−1
108.26.1g2 C 1 −1 1 −1 1 ζ3 ζ3−1 −1 −ζ3−1 −ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3 1 ζ3 ζ3−1 −1 −ζ3−1 −ζ3
108.26.1h1 C 1 1 −1 −1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 1 ζ3−1 ζ3 1 ζ3 ζ3−1
108.26.1h2 C 1 1 −1 −1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 1 ζ3 ζ3−1 1 ζ3−1 ζ3
108.26.2a R 2 2 0 0 2 2 2 2 2 2 0 0 0 0 −1 −1 −1 −1 −1 −1
108.26.2b R 2 −2 0 0 2 2 2 −2 −2 −2 0 0 0 0 −1 −1 −1 1 1 1
108.26.2c1 C 2 2 0 0 2 2ζ3−1 2ζ3 2 2ζ3 2ζ3−1 0 0 0 0 −1 −ζ3−1 −ζ3 −1 −ζ3 −ζ3−1
108.26.2c2 C 2 2 0 0 2 2ζ3 2ζ3−1 2 2ζ3−1 2ζ3 0 0 0 0 −1 −ζ3 −ζ3−1 −1 −ζ3−1 −ζ3
108.26.2d1 C 2 −2 0 0 2 2ζ3−1 2ζ3 −2 −2ζ3 −2ζ3−1 0 0 0 0 −1 −ζ3−1 −ζ3 1 ζ3 ζ3−1
108.26.2d2 C 2 −2 0 0 2 2ζ3 2ζ3−1 −2 −2ζ3−1 −2ζ3 0 0 0 0 −1 −ζ3 −ζ3−1 1 ζ3−1 ζ3
108.26.6a R 6 6 0 0 −3 0 0 −3 0 0 0 0 0 0 0 0 0 0 0 0
108.26.6b R 6 −6 0 0 −3 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed