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Properties

Label 27T399
27T399 1 6 1->6 7 1->7 2 4 2->4 9 2->9 3 5 3->5 8 3->8 4->3 10 4->10 5->1 12 5->12 6->2 11 6->11 7->9 16 7->16 8->7 18 8->18 9->8 17 9->17 15 10->15 19 10->19 13 11->13 21 11->21 14 12->14 20 12->20 13->12 22 13->22 14->10 24 14->24 15->11 23 15->23 16->18 25 16->25 17->16 27 17->27 18->17 26 18->26 19->1 19->21 20->3 20->19 21->2 21->20 22->4 22->24 23->6 23->22 24->5 24->23 25->13 25->27 26->15 26->25 27->14 27->26
Degree $27$
Order $1944$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times C_3^3:A_4$

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Copy content magma:G := TransitiveGroup(27, 399);
 

Group invariants

Abstract group:  $S_3\times C_3^3:A_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $1944=2^{3} \cdot 3^{5}$
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 $t$:  $399$
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:  $(1,6,2,4,3,5)(7,9,8)(10,15,11,13,12,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)$, $(1,7,16,25,13,22,4,10,19)(2,9,17,27,14,24,5,12,20,3,8,18,26,15,23,6,11,21)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$12$:  $A_4$
$18$:  $S_3\times C_3$
$24$:  $A_4\times C_2$
$72$:  12T43
$324$:  $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$
$648$:  18T199

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$, $S_3$

Degree 9: $S_3\times C_3$, $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$

Low degree siblings

18T347, 24T4982 x 2, 24T4985, 36T2770 x 2, 36T2771, 36T2844, 36T2973, 36T2974 x 2

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 $2^{9},1^{9}$ $3$ $2$ $9$ $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)$
2B $2^{6},1^{15}$ $27$ $2$ $6$ $( 7,13)( 8,14)( 9,15)(16,19)(17,20)(18,21)$
2C $2^{11},1^{5}$ $81$ $2$ $11$ $( 1, 2)( 4, 5)( 7,11)( 8,10)( 9,12)(13,14)(16,17)(19,23)(20,22)(21,24)(25,26)$
3A $3^{9}$ $2$ $3$ $18$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)$
3B1 $3^{9}$ $4$ $3$ $18$ $( 1,25, 4)( 2,26, 5)( 3,27, 6)( 7,13,10)( 8,14,11)( 9,15,12)(16,19,22)(17,20,23)(18,21,24)$
3B-1 $3^{9}$ $4$ $3$ $18$ $( 1, 4,25)( 2, 5,26)( 3, 6,27)( 7,10,13)( 8,11,14)( 9,12,15)(16,22,19)(17,23,20)(18,24,21)$
3C $3^{3},1^{18}$ $6$ $3$ $6$ $(16,22,19)(17,23,20)(18,24,21)$
3D1 $3^{9}$ $8$ $3$ $18$ $( 1, 5,27)( 2, 6,25)( 3, 4,26)( 7,11,15)( 8,12,13)( 9,10,14)(16,20,24)(17,21,22)(18,19,23)$
3D-1 $3^{9}$ $8$ $3$ $18$ $( 1, 5,27)( 2, 6,25)( 3, 4,26)( 7,11,15)( 8,12,13)( 9,10,14)(16,23,21)(17,24,19)(18,22,20)$
3E $3^{6},1^{9}$ $12$ $3$ $12$ $( 7,13,10)( 8,14,11)( 9,15,12)(16,22,19)(17,23,20)(18,24,21)$
3F $3^{9}$ $12$ $3$ $18$ $( 1,27, 5)( 2,25, 6)( 3,26, 4)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)$
3G $3^{9}$ $24$ $3$ $18$ $( 1, 2, 3)( 4, 5, 6)( 7,11,15)( 8,12,13)( 9,10,14)(16,20,24)(17,21,22)(18,19,23)(25,26,27)$
3H1 $3^{9}$ $36$ $3$ $18$ $( 1,10,22)( 2,11,23)( 3,12,24)( 4,13,19)( 5,14,20)( 6,15,21)( 7,16,25)( 8,17,26)( 9,18,27)$
3H-1 $3^{9}$ $36$ $3$ $18$ $( 1,22,10)( 2,23,11)( 3,24,12)( 4,19,13)( 5,20,14)( 6,21,15)( 7,25,16)( 8,26,17)( 9,27,18)$
3I1 $3^{9}$ $72$ $3$ $18$ $( 1,18,14)( 2,16,15)( 3,17,13)( 4,21,11)( 5,19,12)( 6,20,10)( 7,27,23)( 8,25,24)( 9,26,22)$
3I-1 $3^{9}$ $72$ $3$ $18$ $( 1,14,18)( 2,15,16)( 3,13,17)( 4,11,21)( 5,12,19)( 6,10,20)( 7,23,27)( 8,24,25)( 9,22,26)$
6A1 $6^{3},3^{3}$ $12$ $6$ $21$ $( 1, 6,25, 3, 4,27)( 2, 5,26)( 7,12,13, 9,10,15)( 8,11,14)(16,24,19,18,22,21)(17,23,20)$
6A-1 $6^{3},3^{3}$ $12$ $6$ $21$ $( 1,27, 4, 3,25, 6)( 2,26, 5)( 7,15,10, 9,13,12)( 8,14,11)(16,21,22,18,19,24)(17,20,23)$
6B $6,3,2^{6},1^{6}$ $18$ $6$ $13$ $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(16,19,22)(17,21,23,18,20,24)(26,27)$
6C $6^{2},3^{2},2^{3},1^{3}$ $36$ $6$ $17$ $( 2, 3)( 5, 6)( 7,10,13)( 8,12,14, 9,11,15)(16,19,22)(17,21,23,18,20,24)(26,27)$
6D $6^{2},3^{5}$ $54$ $6$ $20$ $( 1, 3, 2)( 4, 6, 5)( 7,15, 8,13, 9,14)(10,12,11)(16,21,17,19,18,20)(22,24,23)(25,27,26)$
6E $3^{3},2^{6},1^{6}$ $54$ $6$ $12$ $( 1,25, 4)( 2,26, 5)( 3,27, 6)(10,13)(11,14)(12,15)(16,22)(17,23)(18,24)$
6F $6^{2},3^{5}$ $108$ $6$ $20$ $( 1, 5,27)( 2, 6,25)( 3, 4,26)( 7,14, 9,13, 8,15)(10,11,12)(16,17,18)(19,23,21,22,20,24)$
6G1 $6^{3},3^{3}$ $108$ $6$ $21$ $( 1,24,10, 3,22,12)( 2,23,11)( 4,21,13, 6,19,15)( 5,20,14)( 7,27,16, 9,25,18)( 8,26,17)$
6G-1 $6^{3},3^{3}$ $108$ $6$ $21$ $( 1,12,22, 3,10,24)( 2,11,23)( 4,15,19, 6,13,21)( 5,14,20)( 7,18,25, 9,16,27)( 8,17,26)$
6H $6,3,2^{8},1^{2}$ $162$ $6$ $15$ $( 1, 5,25, 2, 4,26)( 3, 6,27)( 7,11)( 8,10)( 9,12)(13,14)(16,17)(19,23)(20,22)(21,24)$
9A1 $9^{3}$ $36$ $9$ $24$ $( 1,10,19,25, 7,22, 4,13,16)( 2,11,20,26, 8,23, 5,14,17)( 3,12,21,27, 9,24, 6,15,18)$
9A-1 $9^{3}$ $36$ $9$ $24$ $( 1,16,13, 4,22, 7,25,19,10)( 2,17,14, 5,23, 8,26,20,11)( 3,18,15, 6,24, 9,27,21,12)$
9B1 $9^{3}$ $36$ $9$ $24$ $( 1,19, 7,25,16,10, 4,22,13)( 2,20, 8,26,17,11, 5,23,14)( 3,21, 9,27,18,12, 6,24,15)$
9B-1 $9^{3}$ $36$ $9$ $24$ $( 1,13,22, 4,10,16,25, 7,19)( 2,14,23, 5,11,17,26, 8,20)( 3,15,24, 6,12,18,27, 9,21)$
9C1 $9^{3}$ $72$ $9$ $24$ $( 1,24,11,25,18, 8, 4,21,14)( 2,22,12,26,16, 9, 5,19,15)( 3,23,10,27,17, 7, 6,20,13)$
9C-1 $9^{3}$ $72$ $9$ $24$ $( 1,14,21, 4, 8,18,25,11,24)( 2,15,19, 5, 9,16,26,12,22)( 3,13,20, 6, 7,17,27,10,23)$
9D1 $9^{3}$ $72$ $9$ $24$ $( 1,18,11, 4,21, 8,25,24,14)( 2,16,12, 5,19, 9,26,22,15)( 3,17,10, 6,20, 7,27,23,13)$
9D-1 $9^{3}$ $72$ $9$ $24$ $( 1,14,24,25, 8,21, 4,11,18)( 2,15,22,26, 9,19, 5,12,16)( 3,13,23,27, 7,20, 6,10,17)$
18A1 $18,9$ $108$ $18$ $25$ $( 1,24,10, 6,19,15,25,18, 7, 3,22,12, 4,21,13,27,16, 9)( 2,23,11, 5,20,14,26,17, 8)$
18A-1 $18,9$ $108$ $18$ $25$ $( 1, 9,16,27,13,21, 4,12,22, 3, 7,18,25,15,19, 6,10,24)( 2, 8,17,26,14,20, 5,11,23)$
18B1 $18,9$ $108$ $18$ $25$ $( 1,11,19, 5, 7,23,25,14,16, 2,10,20, 4, 8,22,26,13,17)( 3,12,21, 6, 9,24,27,15,18)$
18B-1 $18,9$ $108$ $18$ $25$ $( 1,17,13,26,22, 8, 4,20,10, 2,16,14,25,23, 7, 5,19,11)( 3,18,15,27,24, 9, 6,21,12)$

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

39 x 39 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed