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Properties

Label 18T177
18T177 1 4 1->4 17 1->17 2 3 2->3 18 2->18 6 3->6 14 3->14 5 4->5 13 4->13 5->1 16 5->16 6->2 15 6->15 7 7->2 9 7->9 8 8->1 10 8->10 9->4 11 9->11 10->3 12 10->12 11->6 11->7 12->5 12->8 13->11 13->16 14->12 14->15 15->8 15->17 16->7 16->18 17->9 17->14 18->10 18->13
Degree $18$
Order $576$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^6:C_9$

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Copy content magma:G := TransitiveGroup(18, 177);
 

Group invariants

Abstract group:  $C_2^6:C_9$
Copy content magma:IdentifyGroup(G);
 
Order:  $576=2^{6} \cdot 3^{2}$
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 $n$:  $18$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $177$
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:  $(1,17,9,4,13,11,6,15,8)(2,18,10,3,14,12,5,16,7)$, $(1,4,5)(2,3,6)(7,9,11)(8,10,12)(13,16,18)(14,15,17)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$
$9$:  $C_9$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: None

Degree 9: $C_9$

Low degree siblings

12T166 x 7, 18T177 x 6, 36T954 x 7, 36T955 x 21, 36T956 x 7, 36T957 x 42

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 $2^{6},1^{6}$ $9$ $2$ $6$ $( 1, 2)( 3, 4)( 7, 8)(11,12)(15,16)(17,18)$
2B $2^{4},1^{10}$ $9$ $2$ $4$ $( 9,10)(11,12)(13,14)(17,18)$
2C $2^{6},1^{6}$ $9$ $2$ $6$ $( 3, 4)( 5, 6)( 7, 8)(11,12)(13,14)(15,16)$
2D $2^{4},1^{10}$ $9$ $2$ $4$ $( 3, 4)( 5, 6)(15,16)(17,18)$
2E $2^{4},1^{10}$ $9$ $2$ $4$ $( 1, 2)( 5, 6)(13,14)(15,16)$
2F $2^{2},1^{14}$ $9$ $2$ $2$ $(13,14)(17,18)$
2G $2^{6},1^{6}$ $9$ $2$ $6$ $( 3, 4)( 5, 6)( 7, 8)( 9,10)(15,16)(17,18)$
3A1 $3^{6}$ $64$ $3$ $12$ $( 1, 4, 6)( 2, 3, 5)( 7, 9,12)( 8,10,11)(13,16,17)(14,15,18)$
3A-1 $3^{6}$ $64$ $3$ $12$ $( 1, 6, 4)( 2, 5, 3)( 7,12, 9)( 8,11,10)(13,17,16)(14,18,15)$
9A1 $9^{2}$ $64$ $9$ $16$ $( 1,18, 9, 4,14,12, 6,15, 7)( 2,17,10, 3,13,11, 5,16, 8)$
9A-1 $9^{2}$ $64$ $9$ $16$ $( 1, 7,15, 6,12,14, 4, 9,18)( 2, 8,16, 5,11,13, 3,10,17)$
9A2 $9^{2}$ $64$ $9$ $16$ $( 1, 9,14, 6, 7,18, 4,12,15)( 2,10,13, 5, 8,17, 3,11,16)$
9A-2 $9^{2}$ $64$ $9$ $16$ $( 1,15,12, 4,18, 7, 6,14, 9)( 2,16,11, 3,17, 8, 5,13,10)$
9A4 $9^{2}$ $64$ $9$ $16$ $( 1,14, 7, 4,15, 9, 6,18,12)( 2,13, 8, 3,16,10, 5,17,11)$
9A-4 $9^{2}$ $64$ $9$ $16$ $( 1,12,18, 6, 9,15, 4, 7,14)( 2,11,17, 5,10,16, 3, 8,13)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 3A1 3A-1 9A1 9A-1 9A2 9A-2 9A4 9A-4
Size 1 9 9 9 9 9 9 9 64 64 64 64 64 64 64 64
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A-1 3A1 9A2 9A-2 9A4 9A-4 9A-1 9A1
3 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 1A 3A1 3A-1 3A-1 3A1 3A1 3A-1
Type
576.8661.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
576.8661.1b1 C 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1
576.8661.1b2 C 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
576.8661.1c1 C 1 1 1 1 1 1 1 1 ζ9−3 ζ93 ζ9−4 ζ9−2 ζ9 ζ9−1 ζ94 ζ92
576.8661.1c2 C 1 1 1 1 1 1 1 1 ζ93 ζ9−3 ζ94 ζ92 ζ9−1 ζ9 ζ9−4 ζ9−2
576.8661.1c3 C 1 1 1 1 1 1 1 1 ζ9−3 ζ93 ζ92 ζ9 ζ94 ζ9−4 ζ9−2 ζ9−1
576.8661.1c4 C 1 1 1 1 1 1 1 1 ζ93 ζ9−3 ζ9−2 ζ9−1 ζ9−4 ζ94 ζ92 ζ9
576.8661.1c5 C 1 1 1 1 1 1 1 1 ζ9−3 ζ93 ζ9−1 ζ94 ζ9−2 ζ92 ζ9 ζ9−4
576.8661.1c6 C 1 1 1 1 1 1 1 1 ζ93 ζ9−3 ζ9 ζ9−4 ζ92 ζ9−2 ζ9−1 ζ94
576.8661.9a R 9 −3 −3 1 −3 1 1 5 0 0 0 0 0 0 0 0
576.8661.9b R 9 −3 1 −3 1 5 1 −3 0 0 0 0 0 0 0 0
576.8661.9c R 9 −3 1 1 5 −3 −3 1 0 0 0 0 0 0 0 0
576.8661.9d R 9 1 −3 1 1 −3 5 −3 0 0 0 0 0 0 0 0
576.8661.9e R 9 1 1 5 −3 1 −3 −3 0 0 0 0 0 0 0 0
576.8661.9f R 9 1 5 −3 −3 −3 1 1 0 0 0 0 0 0 0 0
576.8661.9g R 9 5 −3 −3 1 1 −3 1 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed