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Properties

Label 27T96
27T96 1 10 1->10 22 1->22 2 11 2->11 23 2->23 3 12 3->12 24 3->24 4 14 4->14 25 4->25 5 15 5->15 26 5->26 6 13 6->13 27 6->27 7 18 7->18 19 7->19 8 16 8->16 20 8->20 9 17 9->17 21 9->21 10->4 10->11 10->19 11->5 11->12 11->20 12->6 12->10 12->21 13->7 13->14 13->24 14->8 14->15 14->22 15->9 15->13 15->23 16->1 16->17 16->26 17->2 17->18 17->27 18->3 18->16 18->25 19->1 19->13 19->21 20->2 20->14 20->19 21->3 21->15 21->20 22->4 22->16 22->24 23->5 23->17 23->22 24->6 24->18 24->23 25->7 25->10 25->27 26->8 26->11 26->25 27->9 27->12 27->26
Degree $27$
Order $243$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $\He_3:C_3^2$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(27, 96);
 

Group invariants

Abstract group:  $\He_3:C_3^2$
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 $t$:  $96$
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:  $(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26)$, $(1,10,19)(2,11,20)(3,12,21)(4,14,22)(5,15,23)(6,13,24)(7,18,25)(8,16,26)(9,17,27)$, $(1,22,16)(2,23,17)(3,24,18)(4,25,10)(5,26,11)(6,27,12)(7,19,13)(8,20,14)(9,21,15)$
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

27T97 x 9

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{27}$ $1$ $1$ $0$ $()$
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 $3^{6},1^{9}$ $3$ $3$ $12$ $( 4, 5, 6)( 7, 9, 8)(10,12,11)(16,17,18)(19,20,21)(22,24,23)$
3B-1 $3^{6},1^{9}$ $3$ $3$ $12$ $( 4, 6, 5)( 7, 8, 9)(10,11,12)(16,18,17)(19,21,20)(22,23,24)$
3C1 $3^{6},1^{9}$ $3$ $3$ $12$ $(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26)$
3C-1 $3^{6},1^{9}$ $3$ $3$ $12$ $(10,12,11)(13,15,14)(16,18,17)(19,20,21)(22,23,24)(25,26,27)$
3D1 $3^{6},1^{9}$ $3$ $3$ $12$ $( 4, 5, 6)( 7, 9, 8)(13,14,15)(16,18,17)(22,23,24)(25,27,26)$
3D-1 $3^{6},1^{9}$ $3$ $3$ $12$ $( 4, 6, 5)( 7, 8, 9)(13,15,14)(16,17,18)(22,24,23)(25,26,27)$
3E1 $3^{6},1^{9}$ $3$ $3$ $12$ $( 4, 6, 5)( 7, 8, 9)(10,12,11)(13,14,15)(19,20,21)(25,27,26)$
3E-1 $3^{6},1^{9}$ $3$ $3$ $12$ $( 4, 5, 6)( 7, 9, 8)(10,11,12)(13,15,14)(19,21,20)(25,26,27)$
3F1 $3^{9}$ $9$ $3$ $18$ $( 1, 4, 7)( 2, 5, 8)( 3, 6, 9)(10,14,16)(11,15,17)(12,13,18)(19,24,25)(20,22,26)(21,23,27)$
3F-1 $3^{9}$ $9$ $3$ $18$ $( 1, 7, 4)( 2, 8, 5)( 3, 9, 6)(10,16,14)(11,17,15)(12,18,13)(19,25,24)(20,26,22)(21,27,23)$
3G1 $3^{9}$ $9$ $3$ $18$ $( 1,21,12)( 2,19,10)( 3,20,11)( 4,24,13)( 5,22,14)( 6,23,15)( 7,27,17)( 8,25,18)( 9,26,16)$
3G-1 $3^{9}$ $9$ $3$ $18$ $( 1,10,19)( 2,11,20)( 3,12,21)( 4,15,24)( 5,13,22)( 6,14,23)( 7,17,26)( 8,18,27)( 9,16,25)$
3H1 $3^{9}$ $9$ $3$ $18$ $( 1, 4, 7)( 2, 5, 8)( 3, 6, 9)(10,15,18)(11,13,16)(12,14,17)(19,23,26)(20,24,27)(21,22,25)$
3H-1 $3^{9}$ $9$ $3$ $18$ $( 1, 7, 4)( 2, 8, 5)( 3, 9, 6)(10,18,15)(11,16,13)(12,17,14)(19,26,23)(20,27,24)(21,25,22)$
3I1 $3^{9}$ $9$ $3$ $18$ $( 1,17,24)( 2,18,22)( 3,16,23)( 4,10,25)( 5,11,26)( 6,12,27)( 7,15,20)( 8,13,21)( 9,14,19)$
3I-1 $3^{9}$ $9$ $3$ $18$ $( 1,24,18)( 2,22,16)( 3,23,17)( 4,27,12)( 5,25,10)( 6,26,11)( 7,21,15)( 8,19,13)( 9,20,14)$
3J1 $3^{9}$ $9$ $3$ $18$ $( 1, 7, 4)( 2, 8, 5)( 3, 9, 6)(10,17,13)(11,18,14)(12,16,15)(19,27,22)(20,25,23)(21,26,24)$
3J-1 $3^{9}$ $9$ $3$ $18$ $( 1, 4, 7)( 2, 5, 8)( 3, 6, 9)(10,13,17)(11,14,18)(12,15,16)(19,22,27)(20,23,25)(21,24,26)$
3K1 $3^{9}$ $9$ $3$ $18$ $( 1,21,11)( 2,19,12)( 3,20,10)( 4,24,15)( 5,22,13)( 6,23,14)( 7,27,16)( 8,25,17)( 9,26,18)$
3K-1 $3^{9}$ $9$ $3$ $18$ $( 1,10,21)( 2,11,19)( 3,12,20)( 4,15,23)( 5,13,24)( 6,14,22)( 7,17,25)( 8,18,26)( 9,16,27)$
3L1 $3^{9}$ $9$ $3$ $18$ $( 1,17,22)( 2,18,23)( 3,16,24)( 4,10,26)( 5,11,27)( 6,12,25)( 7,15,21)( 8,13,19)( 9,14,20)$
3L-1 $3^{9}$ $9$ $3$ $18$ $( 1,24,16)( 2,22,17)( 3,23,18)( 4,27,10)( 5,25,11)( 6,26,12)( 7,21,13)( 8,19,14)( 9,20,15)$
3M1 $3^{9}$ $9$ $3$ $18$ $( 1,24,17)( 2,22,18)( 3,23,16)( 4,27,11)( 5,25,12)( 6,26,10)( 7,21,14)( 8,19,15)( 9,20,13)$
3M-1 $3^{9}$ $9$ $3$ $18$ $( 1,17,23)( 2,18,24)( 3,16,22)( 4,10,27)( 5,11,25)( 6,12,26)( 7,15,19)( 8,13,20)( 9,14,21)$
3N1 $3^{9}$ $9$ $3$ $18$ $( 1,21,10)( 2,19,11)( 3,20,12)( 4,24,14)( 5,22,15)( 6,23,13)( 7,27,18)( 8,25,16)( 9,26,17)$
3N-1 $3^{9}$ $9$ $3$ $18$ $( 1,10,20)( 2,11,21)( 3,12,19)( 4,15,22)( 5,13,23)( 6,14,24)( 7,17,27)( 8,18,25)( 9,16,26)$
9A1 $9^{3}$ $9$ $9$ $24$ $( 1,27,15, 3,26,14, 2,25,13)( 4,21,17, 6,20,16, 5,19,18)( 7,24,10, 9,23,12, 8,22,11)$
9A-1 $9^{3}$ $9$ $9$ $24$ $( 1,15,26, 2,13,27, 3,14,25)( 4,17,20, 5,18,21, 6,16,19)( 7,10,23, 8,11,24, 9,12,22)$
9B1 $9^{3}$ $9$ $9$ $24$ $( 1,27,14, 3,26,13, 2,25,15)( 4,21,16, 6,20,18, 5,19,17)( 7,24,12, 9,23,11, 8,22,10)$
9B-1 $9^{3}$ $9$ $9$ $24$ $( 1,15,25, 2,13,26, 3,14,27)( 4,17,19, 5,18,20, 6,16,21)( 7,10,22, 8,11,23, 9,12,24)$
9C1 $9^{3}$ $9$ $9$ $24$ $( 1,15,27, 2,13,25, 3,14,26)( 4,17,21, 5,18,19, 6,16,20)( 7,10,24, 8,11,22, 9,12,23)$
9C-1 $9^{3}$ $9$ $9$ $24$ $( 1,27,13, 3,26,15, 2,25,14)( 4,21,18, 6,20,17, 5,19,16)( 7,24,11, 9,23,10, 8,22,12)$

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