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Properties

Label 18T576
18T576 1 11 1->11 15 1->15 2 10 2->10 13 2->13 3 12 3->12 14 3->14 4 8 4->8 4->15 18 4->18 5 7 5->7 5->14 16 5->16 6 9 6->9 6->13 17 6->17 7->10 7->17 8->12 8->16 9->11 9->18 10->4 10->6 10->11 11->5 11->5 11->12 12->4 12->6 12->10 13->3 13->18 14->1 14->1 14->17 15->2 15->3 15->16 16->8 16->13 17->9 17->15 18->7 18->14
Degree $18$
Order $11664$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^4:(C_6^2:C_4)$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(18, 576);
 

Group invariants

Abstract group:  $C_3^4:(C_6^2:C_4)$
Copy content magma:IdentifyGroup(G);
 
Order:  $11664=2^{4} \cdot 3^{6}$
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 $t$:  $576$
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:  $(1,15,3,14)(2,13)(4,18,7,10,6,17,9,11,5,16,8,12)$, $(1,11,5,14)(2,10,4,15)(3,12,6,13)(7,17)(8,16)(9,18)$, $(4,8)(5,7)(6,9)(10,11,12)(13,18,14,17,15,16)$
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, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$36$:  $C_3^2:C_4$
$72$:  $C_3^2:D_4$, 12T40
$144$:  12T79, 12T82
$1296$:  12T209

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $C_3^2:C_4$

Degree 9: None

Low degree siblings

18T578, 36T9252, 36T9253, 36T9254, 36T9255, 36T9256, 36T9257, 36T9258, 36T9266, 36T9267, 36T9268, 36T9269, 36T9270, 36T9271, 36T9272, 36T9398, 36T9443, 36T9501, 36T9522, 36T9557

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 $2^{3},1^{12}$ $18$ $2$ $3$ $(11,12)(14,15)(17,18)$
2B $2^{8},1^{2}$ $81$ $2$ $8$ $( 1, 8)( 2, 7)( 3, 9)( 4, 6)(11,12)(13,16)(14,17)(15,18)$
2C $2^{6},1^{6}$ $81$ $2$ $6$ $( 1, 4)( 2, 6)( 3, 5)(10,13)(11,15)(12,14)$
2D $2^{6},1^{6}$ $81$ $2$ $6$ $( 1, 3)( 4, 6)( 8, 9)(10,12)(13,15)(17,18)$
2E $2^{7},1^{4}$ $162$ $2$ $7$ $( 1, 4)( 2, 5)( 3, 6)( 8, 9)(13,18)(14,17)(15,16)$
3A $3^{3},1^{9}$ $4$ $3$ $6$ $(10,12,11)(13,14,15)(16,18,17)$
3B $3^{6}$ $4$ $3$ $12$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,15,14)(16,17,18)$
3C $3^{3},1^{9}$ $12$ $3$ $6$ $(10,16,13)(11,17,15)(12,18,14)$
3D $3^{2},1^{12}$ $12$ $3$ $4$ $(4,6,5)(7,9,8)$
3E $3^{6}$ $24$ $3$ $12$ $( 1, 3, 2)( 4, 5, 6)( 7, 9, 8)(10,16,13)(11,17,15)(12,18,14)$
3F $3^{3},1^{9}$ $24$ $3$ $6$ $(1,9,5)(2,7,4)(3,8,6)$
3G $3^{5},1^{3}$ $24$ $3$ $10$ $( 1, 3, 2)( 4, 5, 6)( 7, 9, 8)(10,12,11)(16,17,18)$
3H $3^{6}$ $36$ $3$ $12$ $( 1, 7, 4)( 2, 8, 6)( 3, 9, 5)(10,16,13)(11,17,15)(12,18,14)$
3I $3^{4},1^{6}$ $36$ $3$ $8$ $( 1, 3, 2)( 7, 8, 9)(13,14,15)(16,17,18)$
3J $3^{6}$ $48$ $3$ $12$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,13,17)(11,15,18)(12,14,16)$
3K $3^{6}$ $72$ $3$ $12$ $( 1, 5, 8)( 2, 4, 9)( 3, 6, 7)(10,14,16)(11,13,17)(12,15,18)$
3L $3^{6}$ $72$ $3$ $12$ $( 1, 5, 8)( 2, 4, 9)( 3, 6, 7)(10,17,15)(11,18,14)(12,16,13)$
3M $3^{5},1^{3}$ $72$ $3$ $10$ $( 4, 6, 5)( 7, 9, 8)(10,14,17)(11,13,18)(12,15,16)$
3N $3^{5},1^{3}$ $144$ $3$ $10$ $( 1, 7, 5)( 2, 8, 4)( 3, 9, 6)(13,14,15)(16,17,18)$
3O $3^{6}$ $144$ $3$ $12$ $( 1, 5, 9)( 2, 4, 7)( 3, 6, 8)(10,15,16)(11,14,17)(12,13,18)$
4A1 $4^{4},2$ $486$ $4$ $13$ $( 1,17, 8,14)( 2,16, 7,13)( 3,18, 9,15)( 4,12, 6,11)( 5,10)$
4A-1 $4^{4},2$ $486$ $4$ $13$ $( 1,14, 8,17)( 2,13, 7,16)( 3,15, 9,18)( 4,11, 6,12)( 5,10)$
4B1 $4^{3},2^{3}$ $486$ $4$ $12$ $( 1,10, 4,13)( 2,11, 6,15)( 3,12, 5,14)( 7,16)( 8,17)( 9,18)$
4B-1 $4^{3},2^{3}$ $486$ $4$ $12$ $( 1,13, 4,10)( 2,15, 6,11)( 3,14, 5,12)( 7,16)( 8,17)( 9,18)$
6A $6,3,1^{9}$ $36$ $6$ $7$ $(10,13,16)(11,14,17,12,15,18)$
6B $3^{3},2^{3},1^{3}$ $36$ $6$ $9$ $( 2, 3)( 5, 6)( 8, 9)(10,11,12)(13,15,14)(16,17,18)$
6C $6,3^{4}$ $72$ $6$ $13$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,13,16)(11,14,17,12,15,18)$
6D $3^{2},2^{3},1^{6}$ $108$ $6$ $7$ $( 4, 5, 6)( 7, 8, 9)(11,12)(14,15)(17,18)$
6E $3^{3},2^{3},1^{3}$ $108$ $6$ $9$ $( 2, 3)( 5, 6)( 8, 9)(10,13,16)(11,15,17)(12,14,18)$
6F $6,3^{4}$ $108$ $6$ $13$ $( 1, 4, 7)( 2, 5, 8, 3, 6, 9)(10,13,16)(11,15,17)(12,14,18)$
6G $6,3^{4}$ $108$ $6$ $13$ $( 1, 4, 7)( 2, 5, 8, 3, 6, 9)(10,16,13)(11,17,15)(12,18,14)$
6H $3^{3},2^{3},1^{3}$ $216$ $6$ $9$ $( 1, 5, 9)( 2, 4, 7)( 3, 6, 8)(11,12)(13,15)(16,18)$
6I $6,3^{4}$ $216$ $6$ $13$ $( 1, 9, 5, 2, 8, 4)( 3, 7, 6)(10,16,14)(11,17,13)(12,18,15)$
6J $6,3^{4}$ $216$ $6$ $13$ $( 1, 8, 5)( 2, 7, 4, 3, 9, 6)(10,15,17)(11,14,18)(12,13,16)$
6K $6,3^{3},1^{3}$ $216$ $6$ $11$ $( 4, 5, 6)( 7, 8, 9)(10,17,14)(11,16,13,12,18,15)$
6L $6^{2},2^{2},1^{2}$ $324$ $6$ $12$ $( 1, 7, 3, 8, 2, 9)( 4, 6)(11,12)(13,18,14,16,15,17)$
6M $6^{2},3^{2}$ $324$ $6$ $14$ $( 1, 5, 2, 4, 3, 6)( 7, 9, 8)(10,14,11,13,12,15)(16,18,17)$
6N $6,3,2^{3},1^{3}$ $324$ $6$ $10$ $( 1, 6, 2, 5, 3, 4)( 7, 9, 8)(10,15)(11,14)(12,13)$
6O $6^{2},3^{2}$ $324$ $6$ $14$ $( 1, 8, 6, 3, 9, 4)( 2, 7, 5)(10,17,15,12,18,13)(11,16,14)$
6P $6,2^{5},1^{2}$ $324$ $6$ $10$ $( 1, 3)( 4, 8, 5, 9, 6, 7)(10,12)(13,18)(14,16)(15,17)$
6Q $6,3,2^{4},1$ $324$ $6$ $11$ $( 1, 4)( 2, 5)( 3, 6)( 8, 9)(10,11,12)(13,16,14,18,15,17)$
6R $6,2^{4},1^{4}$ $324$ $6$ $9$ $( 4, 8)( 5, 7)( 6, 9)(10,13,12,15,11,14)(16,17)$
6S $6,3,2^{3},1^{3}$ $324$ $6$ $10$ $( 1, 4, 7)( 2, 5, 8, 3, 6, 9)(10,12)(14,15)(16,17)$
6T $6^{2},3,2,1$ $648$ $6$ $13$ $( 1, 5, 3, 6, 2, 4)( 7, 8, 9)(10,17,12,18,11,16)(13,14)$
12A1 $12,4,2$ $972$ $12$ $15$ $( 1,15, 7,17, 3,13, 8,18, 2,14, 9,16)( 4,11, 6,12)( 5,10)$
12A-1 $12,4,2$ $972$ $12$ $15$ $( 1,16, 9,14, 2,18, 8,13, 3,17, 7,15)( 4,12, 6,11)( 5,10)$
12B1 $12,6$ $972$ $12$ $16$ $( 1,15, 5,10, 2,14, 4,11, 3,13, 6,12)( 7,17, 9,16, 8,18)$
12B-1 $12,6$ $972$ $12$ $16$ $( 1,12, 6,13, 3,11, 4,14, 2,10, 5,15)( 7,18, 8,16, 9,17)$

Malle's constant $a(G)$:     $1/3$

Copy content magma:ConjugacyClasses(G);
 

Character table

49 x 49 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed