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Properties

Label 27T111
27T111 1 10 1->10 14 1->14 2 11 2->11 15 2->15 3 12 3->12 13 3->13 4 5 4->5 4->15 16 4->16 6 5->6 5->13 17 5->17 6->4 6->14 18 6->18 7 9 7->9 7->10 7->16 8 8->7 8->11 8->17 9->8 9->12 9->18 20 10->20 22 10->22 21 11->21 23 11->23 19 12->19 24 12->24 13->14 13->24 25 13->25 14->15 14->22 26 14->26 15->13 15->23 27 15->27 16->18 16->20 16->27 17->16 17->21 17->25 18->17 18->19 18->26 19->1 19->4 20->2 20->5 21->3 21->6 22->4 22->8 22->23 23->5 23->9 23->24 24->6 24->7 24->22 25->1 25->9 25->27 26->2 26->7 26->25 27->3 27->8 27->26
Degree $27$
Order $243$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_3^2.C_3^3$

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Copy content magma:G := TransitiveGroup(27, 111);
 

Group invariants

Abstract group:  $C_3^2.C_3^3$
Copy content magma:IdentifyGroup(G);
 
Order:  $243=3^{5}$
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:  $3$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $27$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $111$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $3$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,10,20,2,11,21,3,12,19)(4,15,23,5,13,24,6,14,22)(7,16,27,8,17,25,9,18,26)$, $(1,14,26,2,15,27,3,13,25)(4,16,20,5,17,21,6,18,19)(7,10,22,8,11,23,9,12,24)$, $(4,5,6)(7,9,8)(13,14,15)(16,18,17)(22,23,24)(25,27,26)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$ x 13
$9$:  $C_3^2$ x 13
$27$:  $C_3^2:C_3$ x 3, 27T4
$81$:  27T18

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$ x 4

Degree 9: $C_3^2$

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{27}$ $1$ $1$ $0$ $()$
3A1 $3^{9}$ $1$ $3$ $18$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)$
3A-1 $3^{9}$ $1$ $3$ $18$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)$
3B1 $3^{6},1^{9}$ $3$ $3$ $12$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,11,12)(13,14,15)(16,17,18)$
3B-1 $3^{6},1^{9}$ $3$ $3$ $12$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,18,17)$
3C1 $3^{6},1^{9}$ $3$ $3$ $12$ $( 4, 6, 5)( 7, 8, 9)(13,15,14)(16,17,18)(22,24,23)(25,26,27)$
3C-1 $3^{6},1^{9}$ $3$ $3$ $12$ $( 4, 5, 6)( 7, 9, 8)(13,14,15)(16,18,17)(22,23,24)(25,27,26)$
3D1 $3^{6},1^{9}$ $3$ $3$ $12$ $( 1, 3, 2)( 4, 5, 6)(10,11,12)(16,18,17)(22,24,23)(25,26,27)$
3D-1 $3^{6},1^{9}$ $3$ $3$ $12$ $( 1, 2, 3)( 4, 6, 5)(10,12,11)(16,17,18)(22,23,24)(25,27,26)$
3E1 $3^{6},1^{9}$ $3$ $3$ $12$ $( 1, 2, 3)( 7, 9, 8)(10,12,11)(13,14,15)(22,24,23)(25,26,27)$
3E-1 $3^{6},1^{9}$ $3$ $3$ $12$ $( 1, 3, 2)( 7, 8, 9)(10,11,12)(13,15,14)(22,23,24)(25,27,26)$
9A1 $9^{3}$ $9$ $9$ $24$ $( 1, 6, 9, 3, 5, 8, 2, 4, 7)(10,15,17,12,14,16,11,13,18)(19,24,25,21,23,27,20,22,26)$
9A-1 $9^{3}$ $9$ $9$ $24$ $( 1, 9, 5, 2, 7, 6, 3, 8, 4)(10,17,14,11,18,15,12,16,13)(19,25,23,20,26,24,21,27,22)$
9B1 $9^{3}$ $9$ $9$ $24$ $( 1,21,11, 3,20,10, 2,19,12)( 4,22,15, 6,24,14, 5,23,13)( 7,27,18, 9,26,17, 8,25,16)$
9B-1 $9^{3}$ $9$ $9$ $24$ $( 1,11,20, 2,12,21, 3,10,19)( 4,15,24, 5,13,22, 6,14,23)( 7,18,26, 8,16,27, 9,17,25)$
9C1 $9^{3}$ $9$ $9$ $24$ $( 1,23,18, 3,22,17, 2,24,16)( 4,26,12, 6,25,11, 5,27,10)( 7,20,15, 9,19,14, 8,21,13)$
9C-1 $9^{3}$ $9$ $9$ $24$ $( 1,18,24, 2,16,22, 3,17,23)( 4,12,27, 5,10,25, 6,11,26)( 7,15,21, 8,13,19, 9,14,20)$
9D1 $9^{3}$ $9$ $9$ $24$ $( 1, 6, 8, 3, 5, 7, 2, 4, 9)(10,15,16,12,14,18,11,13,17)(19,24,27,21,23,26,20,22,25)$
9D-1 $9^{3}$ $9$ $9$ $24$ $( 1, 9, 4, 2, 7, 5, 3, 8, 6)(10,17,13,11,18,14,12,16,15)(19,25,22,20,26,23,21,27,24)$
9E1 $9^{3}$ $9$ $9$ $24$ $( 1,21,11, 3,20,10, 2,19,12)( 4,24,13, 6,23,15, 5,22,14)( 7,25,17, 9,27,16, 8,26,18)$
9E-1 $9^{3}$ $9$ $9$ $24$ $( 1,11,20, 2,12,21, 3,10,19)( 4,13,23, 5,14,24, 6,15,22)( 7,17,27, 8,18,25, 9,16,26)$
9F1 $9^{3}$ $9$ $9$ $24$ $( 1,27,15, 3,26,14, 2,25,13)( 4,19,16, 6,21,18, 5,20,17)( 7,22,12, 9,24,11, 8,23,10)$
9F-1 $9^{3}$ $9$ $9$ $24$ $( 1,13,26, 2,14,27, 3,15,25)( 4,17,21, 5,18,19, 6,16,20)( 7,10,24, 8,11,22, 9,12,23)$
9G1 $9^{3}$ $9$ $9$ $24$ $( 1, 9, 6, 2, 7, 4, 3, 8, 5)(10,17,15,11,18,13,12,16,14)(19,25,24,20,26,22,21,27,23)$
9G-1 $9^{3}$ $9$ $9$ $24$ $( 1, 6, 7, 3, 5, 9, 2, 4, 8)(10,15,18,12,14,17,11,13,16)(19,24,26,21,23,25,20,22,27)$
9H1 $9^{3}$ $9$ $9$ $24$ $( 1,11,20, 2,12,21, 3,10,19)( 4,14,22, 5,15,23, 6,13,24)( 7,16,25, 8,17,26, 9,18,27)$
9H-1 $9^{3}$ $9$ $9$ $24$ $( 1,21,11, 3,20,10, 2,19,12)( 4,23,14, 6,22,13, 5,24,15)( 7,26,16, 9,25,18, 8,27,17)$
9I1 $9^{3}$ $9$ $9$ $24$ $( 1,23,17, 3,22,16, 2,24,18)( 4,25,12, 6,27,11, 5,26,10)( 7,21,13, 9,20,15, 8,19,14)$
9I-1 $9^{3}$ $9$ $9$ $24$ $( 1,18,23, 2,16,24, 3,17,22)( 4,10,25, 5,11,26, 6,12,27)( 7,14,21, 8,15,19, 9,13,20)$
9J1 $9^{3}$ $9$ $9$ $24$ $( 1,27,13, 3,26,15, 2,25,14)( 4,21,18, 6,20,17, 5,19,16)( 7,23,12, 9,22,11, 8,24,10)$
9J-1 $9^{3}$ $9$ $9$ $24$ $( 1,13,27, 2,14,25, 3,15,26)( 4,18,21, 5,16,19, 6,17,20)( 7,12,23, 8,10,24, 9,11,22)$
9K1 $9^{3}$ $9$ $9$ $24$ $( 1,13,25, 2,14,26, 3,15,27)( 4,16,21, 5,17,19, 6,18,20)( 7,11,22, 8,12,23, 9,10,24)$
9K-1 $9^{3}$ $9$ $9$ $24$ $( 1,27,14, 3,26,13, 2,25,15)( 4,20,17, 6,19,16, 5,21,18)( 7,24,12, 9,23,11, 8,22,10)$
9L1 $9^{3}$ $9$ $9$ $24$ $( 1,23,16, 3,22,18, 2,24,17)( 4,27,12, 6,26,11, 5,25,10)( 7,19,14, 9,21,13, 8,20,15)$
9L-1 $9^{3}$ $9$ $9$ $24$ $( 1,18,22, 2,16,23, 3,17,24)( 4,11,26, 5,12,27, 6,10,25)( 7,13,21, 8,14,19, 9,15,20)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

35 x 35 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed