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Properties

Label 27T466
27T466 1 13 1->13 20 1->20 23 1->23 2 14 2->14 21 2->21 22 2->22 3 15 3->15 19 3->19 24 3->24 4 18 4->18 4->24 25 4->25 5 16 5->16 5->22 27 5->27 6 17 6->17 6->23 26 6->26 7 11 7->11 7->21 7->25 8 12 8->12 8->20 8->26 9 10 9->10 9->19 9->27 10->6 10->6 10->12 11->4 11->4 12->5 12->5 13->7 13->8 13->14 14->8 14->9 15->7 15->9 16->1 16->2 17->2 17->3 17->18 18->1 18->3 19->7 19->16 19->23 20->9 20->17 20->24 21->8 21->18 21->22 22->3 22->11 22->25 23->2 23->12 23->26 24->1 24->10 24->27 25->5 25->15 25->21 26->4 26->13 26->19 27->6 27->14 27->20
Degree $27$
Order $2916$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^4:S_3^2$

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Copy content magma:G := TransitiveGroup(27, 466);
 

Group invariants

Abstract group:  $C_3^4:S_3^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $2916=2^{2} \cdot 3^{6}$
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 $n$:  $27$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $466$
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:  $(1,20,17,3,19,16,2,21,18)(4,24,10,6,23,12,5,22,11)(7,25,15,9,27,14,8,26,13)$, $(1,13,8,12,5,16)(2,14,9,10,6,17)(3,15,7,11,4,18)(19,23,26)(20,24,27)(21,22,25)$, $(1,23,2,22,3,24)(4,25,5,27,6,26)(7,21,8,20,9,19)(10,12)(13,14)(17,18)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 5
$12$:  $D_{6}$ x 5
$18$:  $C_3^2:C_2$
$36$:  $S_3^2$ x 4, 18T12
$54$:  $(C_3^2:C_3):C_2$
$108$:  $C_3^2 : D_{6} $, 18T52, 18T58
$324$:  18T133, 18T135
$972$:  18T244

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 9: $(C_3^2:C_3):C_2$

Low degree siblings

27T471, 27T494

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{27}$ $1$ $1$ $0$ $()$
2A $2^{9},1^{9}$ $27$ $2$ $9$ $( 1, 3)( 4, 5)( 8, 9)(11,12)(13,15)(16,17)(19,20)(23,24)(25,27)$
2B $2^{9},1^{9}$ $27$ $2$ $9$ $(10,27)(11,25)(12,26)(13,21)(14,19)(15,20)(16,24)(17,22)(18,23)$
2C $2^{12},1^{3}$ $81$ $2$ $12$ $( 1, 2)( 4, 5)( 7, 8)(10,23)(11,22)(12,24)(13,26)(14,25)(15,27)(16,20)(17,19)(18,21)$
3A $3^{9}$ $2$ $3$ $18$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)$
3B1 $3^{9}$ $3$ $3$ $18$ $( 1, 9, 5)( 2, 7, 6)( 3, 8, 4)(10,18,14)(11,16,15)(12,17,13)(19,27,23)(20,25,24)(21,26,22)$
3B-1 $3^{9}$ $3$ $3$ $18$ $( 1, 5, 9)( 2, 6, 7)( 3, 4, 8)(10,14,18)(11,15,16)(12,13,17)(19,23,27)(20,24,25)(21,22,26)$
3C $3^{6},1^{9}$ $6$ $3$ $12$ $(10,12,11)(13,15,14)(16,18,17)(19,20,21)(22,23,24)(25,26,27)$
3D1 $3^{9}$ $6$ $3$ $18$ $( 1, 7, 4)( 2, 8, 5)( 3, 9, 6)(10,17,15)(11,18,13)(12,16,14)(19,27,23)(20,25,24)(21,26,22)$
3D-1 $3^{9}$ $6$ $3$ $18$ $( 1, 4, 7)( 2, 5, 8)( 3, 6, 9)(10,15,17)(11,13,18)(12,14,16)(19,23,27)(20,24,25)(21,22,26)$
3E $3^{6},1^{9}$ $18$ $3$ $12$ $( 1, 4, 7)( 2, 5, 8)( 3, 6, 9)(19,25,22)(20,26,23)(21,27,24)$
3F $3^{6},1^{9}$ $18$ $3$ $12$ $( 1, 3, 2)( 4, 5, 6)(10,12,11)(13,14,15)(19,21,20)(22,23,24)$
3G $3^{9}$ $18$ $3$ $18$ $( 1,11,21)( 2,12,19)( 3,10,20)( 4,14,24)( 5,15,22)( 6,13,23)( 7,17,27)( 8,18,25)( 9,16,26)$
3H1 $3^{9}$ $18$ $3$ $18$ $( 1, 5, 7)( 2, 6, 8)( 3, 4, 9)(10,13,17)(11,14,18)(12,15,16)(19,24,27)(20,22,25)(21,23,26)$
3H-1 $3^{9}$ $18$ $3$ $18$ $( 1, 7, 5)( 2, 8, 6)( 3, 9, 4)(10,17,13)(11,18,14)(12,16,15)(19,27,24)(20,25,22)(21,26,23)$
3I $3^{9}$ $36$ $3$ $18$ $( 1,14,27)( 2,15,25)( 3,13,26)( 4,16,19)( 5,17,20)( 6,18,21)( 7,12,23)( 8,10,24)( 9,11,22)$
3J $3^{9}$ $36$ $3$ $18$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,18,14)(11,16,15)(12,17,13)(19,24,26)(20,22,27)(21,23,25)$
3K $3^{9}$ $36$ $3$ $18$ $( 1,10,20)( 2,11,21)( 3,12,19)( 4,13,23)( 5,14,24)( 6,15,22)( 7,16,26)( 8,17,27)( 9,18,25)$
3L1 $3^{9}$ $36$ $3$ $18$ $( 1,13,27)( 2,14,25)( 3,15,26)( 4,18,19)( 5,16,20)( 6,17,21)( 7,11,23)( 8,12,24)( 9,10,22)$
3L-1 $3^{9}$ $36$ $3$ $18$ $( 1,15,27)( 2,13,25)( 3,14,26)( 4,17,19)( 5,18,20)( 6,16,21)( 7,10,23)( 8,11,24)( 9,12,22)$
3M $3^{9}$ $54$ $3$ $18$ $( 1,14,24)( 2,15,22)( 3,13,23)( 4,17,27)( 5,18,25)( 6,16,26)( 7,11,21)( 8,12,19)( 9,10,20)$
3N $3^{9}$ $54$ $3$ $18$ $( 1,15,19)( 2,13,20)( 3,14,21)( 4,18,22)( 5,16,23)( 6,17,24)( 7,12,25)( 8,10,26)( 9,11,27)$
3O $3^{8},1^{3}$ $108$ $3$ $16$ $( 1, 2, 3)( 4, 6, 5)(10,16,15)(11,17,13)(12,18,14)(19,24,27)(20,22,25)(21,23,26)$
6A1 $6^{3},3^{3}$ $27$ $6$ $21$ $( 1, 4, 9, 3, 5, 8)( 2, 6, 7)(10,14,18)(11,13,16,12,15,17)(19,24,27,20,23,25)(21,22,26)$
6A-1 $6^{3},3^{3}$ $27$ $6$ $21$ $( 1, 8, 4, 2, 7, 5)( 3, 9, 6)(10,18,13,12,16,15)(11,17,14)(19,25,22)(20,27,23,21,26,24)$
6B $6^{3},3^{3}$ $54$ $6$ $21$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,25,12,27,11,26)(13,19,15,21,14,20)(16,22,18,24,17,23)$
6C1 $6^{4},3$ $81$ $6$ $22$ $( 1, 8, 4, 2, 7, 5)( 3, 9, 6)(10,20,13,23,16,26)(11,19,14,22,17,25)(12,21,15,24,18,27)$
6C-1 $6^{4},3$ $81$ $6$ $22$ $( 1, 5, 7, 2, 4, 8)( 3, 6, 9)(10,26,16,23,13,20)(11,25,17,22,14,19)(12,27,18,24,15,21)$
6D1 $6^{3},3^{3}$ $81$ $6$ $21$ $( 1,10, 7,16, 4,13)( 2,11, 8,17, 5,14)( 3,12, 9,18, 6,15)(19,22,25)(20,23,26)(21,24,27)$
6D-1 $6^{3},3^{3}$ $81$ $6$ $21$ $( 1,13, 4,16, 7,10)( 2,14, 5,17, 8,11)( 3,15, 6,18, 9,12)(19,25,22)(20,26,23)(21,27,24)$
6E $6^{2},3^{2},2^{3},1^{3}$ $162$ $6$ $17$ $( 1, 7, 4)( 2, 9, 5, 3, 8, 6)(10,12)(13,15)(16,18)(19,23,25,20,22,26)(21,24,27)$
6F $6^{2},3^{2},2^{3},1^{3}$ $162$ $6$ $17$ $( 1,10, 3,12, 2,11)( 4,14, 5,15, 6,13)( 7,18)( 8,16)( 9,17)(19,20,21)(22,24,23)$
6G $6^{3},3^{3}$ $162$ $6$ $21$ $( 1,21,11)( 2,20,12, 3,19,10)( 4,22,14, 5,24,15)( 6,23,13)( 7,26,17, 9,27,16)( 8,25,18)$
6H $6^{3},3^{3}$ $162$ $6$ $21$ $( 1,22,14, 2,24,15)( 3,23,13)( 4,25,17, 5,27,18)( 6,26,16)( 7,19,11, 8,21,12)( 9,20,10)$
6I $6^{3},3^{3}$ $162$ $6$ $21$ $( 1,19,15)( 2,21,13, 3,20,14)( 4,23,18, 5,22,16)( 6,24,17)( 7,27,12, 9,25,11)( 8,26,10)$
6J $6^{3},2^{3},1^{3}$ $162$ $6$ $18$ $( 2, 3)( 4, 6)( 7, 8)(10,26,12,27,11,25)(13,19,15,20,14,21)(16,24,18,22,17,23)$
6K1 $6^{3},3^{3}$ $162$ $6$ $21$ $( 1,12, 5,15, 7,16)( 2,10, 6,13, 8,17)( 3,11, 4,14, 9,18)(19,27,24)(20,25,22)(21,26,23)$
6K-1 $6^{3},3^{3}$ $162$ $6$ $21$ $( 1,16, 7,15, 5,12)( 2,17, 8,13, 6,10)( 3,18, 9,14, 4,11)(19,24,27)(20,22,25)(21,23,26)$
6L1 $6^{4},3$ $162$ $6$ $22$ $( 1,13, 7,11, 4,18)( 2,15, 8,10, 5,17)( 3,14, 9,12, 6,16)(19,24,27,20,23,25)(21,22,26)$
6L-1 $6^{4},3$ $162$ $6$ $22$ $( 1,18, 4,11, 7,13)( 2,17, 5,10, 8,15)( 3,16, 6,12, 9,14)(19,25,23,20,27,24)(21,26,22)$
9A $9^{3}$ $108$ $9$ $24$ $( 1,26,10, 3,25,12, 2,27,11)( 4,21,15, 6,20,14, 5,19,13)( 7,22,17, 9,24,16, 8,23,18)$
9B $9^{3}$ $108$ $9$ $24$ $( 1,12,23, 3,11,22, 2,10,24)( 4,14,27, 6,13,26, 5,15,25)( 7,16,19, 9,18,21, 8,17,20)$

Malle's constant $a(G)$:     $1/9$

Copy content magma:ConjugacyClasses(G);
 

Character table

42 x 42 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed