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Properties

Label 18T368
18T368 1 12 1->12 18 1->18 2 11 2->11 17 2->17 3 3->1 14 3->14 4 4->2 13 4->13 5 5->3 15 5->15 6 6->4 16 6->16 7 7->5 7->17 8 8->6 8->18 9 9->2 9->7 10 10->1 10->8 11->3 11->9 12->4 12->10 13->5 13->12 14->6 14->11 15->7 15->14 16->8 16->13 17->10 17->16 18->9 18->15
Degree $18$
Order $2304$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^8:C_9$

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Copy content magma:G := TransitiveGroup(18, 368);
 

Group invariants

Abstract group:  $C_2^8:C_9$
Copy content magma:IdentifyGroup(G);
 
Order:  $2304=2^{8} \cdot 3^{2}$
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 $t$:  $368$
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:  $(1,18,15,14,11,9,7,5,3)(2,17,16,13,12,10,8,6,4)$, $(1,12,4,13,5,15,7,17,10)(2,11,3,14,6,16,8,18,9)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$
$9$:  $C_9$
$12$:  $A_4$
$36$:  $C_2^2 : C_9$
$576$:  12T166

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: None

Degree 9: $C_9$

Low degree siblings

18T368 x 20, 24T5077 x 21, 36T3083 x 21, 36T3084 x 21, 36T3408 x 21, 36T3409 x 63, 36T3410 x 63, 36T3411 x 126, 36T3412 x 126, 36T3413 x 126, 36T3414 x 126, 36T3462 x 7, 36T3463 x 7, 36T3464 x 63, 36T3465 x 63, 36T3466 x 42, 36T3467 x 126, 36T3468 x 126

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 $2^{6},1^{6}$ $3$ $2$ $6$ $( 3, 4)( 5, 6)( 9,10)(11,12)(15,16)(17,18)$
2B $2^{4},1^{10}$ $9$ $2$ $4$ $( 1, 2)( 7, 8)( 9,10)(15,16)$
2C $2^{2},1^{14}$ $9$ $2$ $2$ $( 3, 4)(13,14)$
2D $2^{6},1^{6}$ $9$ $2$ $6$ $( 5, 6)( 9,10)(11,12)(13,14)(15,16)(17,18)$
2E $2^{6},1^{6}$ $9$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(11,12)(17,18)$
2F $2^{4},1^{10}$ $9$ $2$ $4$ $( 1, 2)( 5, 6)( 7, 8)(17,18)$
2G $2^{6},1^{6}$ $9$ $2$ $6$ $( 3, 4)( 5, 6)( 9,10)(13,14)(15,16)(17,18)$
2H $2^{2},1^{14}$ $9$ $2$ $2$ $(11,12)(13,14)$
2I $2^{6},1^{6}$ $9$ $2$ $6$ $( 1, 2)( 3, 4)( 7, 8)( 9,10)(11,12)(15,16)$
2J $2^{6},1^{6}$ $9$ $2$ $6$ $( 3, 4)( 5, 6)( 7, 8)( 9,10)(13,14)(17,18)$
2K $2^{4},1^{10}$ $9$ $2$ $4$ $( 1, 2)( 5, 6)(15,16)(17,18)$
2L $2^{4},1^{10}$ $9$ $2$ $4$ $( 1, 2)( 3, 4)( 9,10)(11,12)$
2M $2^{4},1^{10}$ $9$ $2$ $4$ $( 7, 8)(11,12)(13,14)(15,16)$
2N $2^{6},1^{6}$ $9$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 6)(13,14)(15,16)(17,18)$
2O $2^{4},1^{10}$ $9$ $2$ $4$ $( 5, 6)( 7, 8)( 9,10)(17,18)$
2P $2^{4},1^{10}$ $9$ $2$ $4$ $( 3, 4)( 7, 8)(11,12)(15,16)$
2Q $2^{4},1^{10}$ $9$ $2$ $4$ $( 1, 2)( 9,10)(11,12)(13,14)$
2R $2^{2},1^{14}$ $9$ $2$ $2$ $( 5, 6)(11,12)$
2S $2^{8},1^{2}$ $9$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2T $2^{4},1^{10}$ $9$ $2$ $4$ $( 1, 2)( 7, 8)(13,14)(17,18)$
2U $2^{4},1^{10}$ $9$ $2$ $4$ $( 3, 4)( 9,10)(15,16)(17,18)$
2V $2^{6},1^{6}$ $9$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 9,10)(11,12)(13,14)$
2W $2^{4},1^{10}$ $9$ $2$ $4$ $( 5, 6)( 7, 8)(11,12)(15,16)$
2X $2^{4},1^{10}$ $9$ $2$ $4$ $( 3, 4)( 7, 8)( 9,10)(17,18)$
2Y $2^{4},1^{10}$ $9$ $2$ $4$ $( 1, 2)(13,14)(15,16)(17,18)$
2Z $2^{4},1^{10}$ $9$ $2$ $4$ $( 1, 2)( 9,10)(13,14)(15,16)$
2AA $2^{2},1^{14}$ $9$ $2$ $2$ $(3,4)(7,8)$
2AB $2^{6},1^{6}$ $9$ $2$ $6$ $( 5, 6)( 7, 8)( 9,10)(11,12)(15,16)(17,18)$
2AC $2^{6},1^{6}$ $9$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 6)(11,12)(13,14)(17,18)$
3A1 $3^{6}$ $64$ $3$ $12$ $( 1,14, 8)( 2,13, 7)( 3,16,10)( 4,15, 9)( 5,17,11)( 6,18,12)$
3A-1 $3^{6}$ $64$ $3$ $12$ $( 1, 8,14)( 2, 7,13)( 3,10,16)( 4, 9,15)( 5,11,17)( 6,12,18)$
6A1 $6^{2},3^{2}$ $192$ $6$ $14$ $( 1, 8,14)( 2, 7,13)( 3,10,16, 4, 9,15)( 5,12,17, 6,11,18)$
6A-1 $6^{2},3^{2}$ $192$ $6$ $14$ $( 1,14, 8)( 2,13, 7)( 3,15, 9, 4,16,10)( 5,18,11, 6,17,12)$
9A1 $9^{2}$ $256$ $9$ $16$ $( 1, 5,10,14,17, 3, 8,11,16)( 2, 6, 9,13,18, 4, 7,12,15)$
9A-1 $9^{2}$ $256$ $9$ $16$ $( 1,16,11, 8, 3,17,14,10, 5)( 2,15,12, 7, 4,18,13, 9, 6)$
9A2 $9^{2}$ $256$ $9$ $16$ $( 1,10,17, 8,16, 5,14, 3,11)( 2, 9,18, 7,15, 6,13, 4,12)$
9A-2 $9^{2}$ $256$ $9$ $16$ $( 1,11, 3,14, 5,16, 8,17,10)( 2,12, 4,13, 6,15, 7,18, 9)$
9A4 $9^{2}$ $256$ $9$ $16$ $( 1,17,16,14,11,10, 8, 5, 3)( 2,18,15,13,12, 9, 7, 6, 4)$
9A-4 $9^{2}$ $256$ $9$ $16$ $( 1, 3, 5, 8,10,11,14,16,17)( 2, 4, 6, 7, 9,12,13,15,18)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

40 x 40 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed