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Properties

Label 36T27
36T27 1 6 1->6 9 1->9 2 5 2->5 10 2->10 3 8 3->8 11 3->11 4 7 4->7 12 4->12 17 5->17 35 5->35 18 6->18 36 6->36 20 7->20 33 7->33 19 8->19 34 8->34 15 9->15 26 9->26 16 10->16 25 10->25 13 11->13 28 11->28 14 12->14 27 12->27 13->18 21 13->21 14->17 22 14->22 15->20 24 15->24 16->19 23 16->23 17->9 32 17->32 18->10 31 18->31 19->12 30 19->30 20->11 29 20->29 21->3 21->27 22->4 22->28 23->2 23->26 24->1 24->25 25->32 25->34 26->31 26->33 27->29 27->35 28->30 28->36 29->6 29->23 30->5 30->24 31->8 31->22 32->7 32->21 33->1 33->16 34->2 34->15 35->4 35->13 36->3 36->14
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times C_{12}$

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Copy content magma:G := TransitiveGroup(36, 27);
 

Group invariants

Abstract group:  $S_3\times C_{12}$
Copy content magma:IdentifyGroup(G);
 
Order:  $72=2^{3} \cdot 3^{2}$
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 $n$:  $36$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $27$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $12$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,9,26,33,16,23,2,10,25,34,15,24)(3,11,28,36,14,22,4,12,27,35,13,21)(5,17,32,7,20,29,6,18,31,8,19,30)$, $(1,6,36,3,8,34,2,5,35,4,7,33)(9,15,20,11,13,18,10,16,19,12,14,17)(21,27,29,23,26,31,22,28,30,24,25,32)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$8$:  $C_4\times C_2$
$12$:  $D_{6}$, $C_{12}$ x 2, $C_6\times C_2$
$18$:  $S_3\times C_3$
$24$:  $S_3 \times C_4$, 24T2
$36$:  $C_6\times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$, $S_3$

Degree 4: $C_4$

Degree 6: $C_6$, $D_{6}$

Degree 9: $S_3\times C_3$

Degree 12: $C_{12}$, $S_3 \times C_4$

Degree 18: $S_3 \times C_6$

Low degree siblings

24T65, 36T27

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{36}$ $1$ $1$ $0$ $()$
2A $2^{18}$ $1$ $2$ $18$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
2B $2^{18}$ $3$ $2$ $18$ $( 1,36)( 2,35)( 3,34)( 4,33)( 5, 6)( 7, 8)( 9,14)(10,13)(11,15)(12,16)(17,18)(19,20)(21,25)(22,26)(23,27)(24,28)(29,30)(31,32)$
2C $2^{12},1^{12}$ $3$ $2$ $12$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)(13,19)(14,20)(15,18)(16,17)(25,29)(26,30)(27,31)(28,32)$
3A1 $3^{12}$ $1$ $3$ $24$ $( 1,16,25)( 2,15,26)( 3,14,27)( 4,13,28)( 5,20,31)( 6,19,32)( 7,18,30)( 8,17,29)( 9,23,34)(10,24,33)(11,22,35)(12,21,36)$
3A-1 $3^{12}$ $1$ $3$ $24$ $( 1,25,16)( 2,26,15)( 3,27,14)( 4,28,13)( 5,31,20)( 6,32,19)( 7,30,18)( 8,29,17)( 9,34,23)(10,33,24)(11,35,22)(12,36,21)$
3B $3^{12}$ $2$ $3$ $24$ $( 1, 8,35)( 2, 7,36)( 3, 5,33)( 4, 6,34)( 9,13,19)(10,14,20)(11,16,17)(12,15,18)(21,26,30)(22,25,29)(23,28,32)(24,27,31)$
3C1 $3^{12}$ $2$ $3$ $24$ $( 1,17,22)( 2,18,21)( 3,20,24)( 4,19,23)( 5,10,27)( 6, 9,28)( 7,12,26)( 8,11,25)(13,32,34)(14,31,33)(15,30,36)(16,29,35)$
3C-1 $3^{12}$ $2$ $3$ $24$ $( 1,22,17)( 2,21,18)( 3,24,20)( 4,23,19)( 5,27,10)( 6,28, 9)( 7,26,12)( 8,25,11)(13,34,32)(14,33,31)(15,36,30)(16,35,29)$
4A1 $4^{9}$ $1$ $4$ $27$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)(17,19,18,20)(21,24,22,23)(25,28,26,27)(29,32,30,31)(33,35,34,36)$
4A-1 $4^{9}$ $1$ $4$ $27$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)(17,20,18,19)(21,23,22,24)(25,27,26,28)(29,31,30,32)(33,36,34,35)$
4B1 $4^{9}$ $3$ $4$ $27$ $( 1, 4, 2, 3)( 5,35, 6,36)( 7,33, 8,34)( 9,18,10,17)(11,19,12,20)(13,15,14,16)(21,31,22,32)(23,30,24,29)(25,28,26,27)$
4B-1 $4^{9}$ $3$ $4$ $27$ $( 1, 3, 2, 4)( 5,36, 6,35)( 7,34, 8,33)( 9,17,10,18)(11,20,12,19)(13,16,14,15)(21,32,22,31)(23,29,24,30)(25,27,26,28)$
6A1 $6^{6}$ $1$ $6$ $30$ $( 1,26,16, 2,25,15)( 3,28,14, 4,27,13)( 5,32,20, 6,31,19)( 7,29,18, 8,30,17)( 9,33,23,10,34,24)(11,36,22,12,35,21)$
6A-1 $6^{6}$ $1$ $6$ $30$ $( 1,15,25, 2,16,26)( 3,13,27, 4,14,28)( 5,19,31, 6,20,32)( 7,17,30, 8,18,29)( 9,24,34,10,23,33)(11,21,35,12,22,36)$
6B $6^{6}$ $2$ $6$ $30$ $( 1,36, 8, 2,35, 7)( 3,34, 5, 4,33, 6)( 9,20,13,10,19,14)(11,18,16,12,17,15)(21,29,26,22,30,25)(23,31,28,24,32,27)$
6C1 $6^{6}$ $2$ $6$ $30$ $( 1,21,17, 2,22,18)( 3,23,20, 4,24,19)( 5,28,10, 6,27, 9)( 7,25,12, 8,26,11)(13,33,32,14,34,31)(15,35,30,16,36,29)$
6C-1 $6^{6}$ $2$ $6$ $30$ $( 1,18,22, 2,17,21)( 3,19,24, 4,20,23)( 5, 9,27, 6,10,28)( 7,11,26, 8,12,25)(13,31,34,14,32,33)(15,29,36,16,30,35)$
6D1 $6^{6}$ $3$ $6$ $30$ $( 1,21,16,36,25,12)( 2,22,15,35,26,11)( 3,23,14,34,27, 9)( 4,24,13,33,28,10)( 5,32,20, 6,31,19)( 7,29,18, 8,30,17)$
6D-1 $6^{6}$ $3$ $6$ $30$ $( 1,12,25,36,16,21)( 2,11,26,35,15,22)( 3, 9,27,34,14,23)( 4,10,28,33,13,24)( 5,19,31, 6,20,32)( 7,17,30, 8,18,29)$
6E1 $6^{4},3^{4}$ $3$ $6$ $28$ $( 1,29,16, 8,25,17)( 2,30,15, 7,26,18)( 3,31,14, 5,27,20)( 4,32,13, 6,28,19)( 9,34,23)(10,33,24)(11,35,22)(12,36,21)$
6E-1 $6^{4},3^{4}$ $3$ $6$ $28$ $( 1,17,25, 8,16,29)( 2,18,26, 7,15,30)( 3,20,27, 5,14,31)( 4,19,28, 6,13,32)( 9,23,34)(10,24,33)(11,22,35)(12,21,36)$
12A1 $12^{3}$ $1$ $12$ $33$ $( 1,14,26, 4,16,27, 2,13,25, 3,15,28)( 5,18,32, 8,20,30, 6,17,31, 7,19,29)( 9,22,33,12,23,35,10,21,34,11,24,36)$
12A-1 $12^{3}$ $1$ $12$ $33$ $( 1,28,15, 3,25,13, 2,27,16, 4,26,14)( 5,29,19, 7,31,17, 6,30,20, 8,32,18)( 9,36,24,11,34,21,10,35,23,12,33,22)$
12A5 $12^{3}$ $1$ $12$ $33$ $( 1,27,15, 4,25,14, 2,28,16, 3,26,13)( 5,30,19, 8,31,18, 6,29,20, 7,32,17)( 9,35,24,12,34,22,10,36,23,11,33,21)$
12A-5 $12^{3}$ $1$ $12$ $33$ $( 1,13,26, 3,16,28, 2,14,25, 4,15,27)( 5,17,32, 7,20,29, 6,18,31, 8,19,30)( 9,21,33,11,23,36,10,22,34,12,24,35)$
12B1 $12^{3}$ $2$ $12$ $33$ $( 1, 6,36, 3, 8,34, 2, 5,35, 4, 7,33)( 9,15,20,11,13,18,10,16,19,12,14,17)(21,27,29,23,26,31,22,28,30,24,25,32)$
12B-1 $12^{3}$ $2$ $12$ $33$ $( 1, 5,36, 4, 8,33, 2, 6,35, 3, 7,34)( 9,16,20,12,13,17,10,15,19,11,14,18)(21,28,29,24,26,32,22,27,30,23,25,31)$
12C1 $12^{3}$ $2$ $12$ $33$ $( 1,19,21, 3,17,23, 2,20,22, 4,18,24)( 5,11,28, 7,10,25, 6,12,27, 8, 9,26)(13,30,33,16,32,36,14,29,34,15,31,35)$
12C-1 $12^{3}$ $2$ $12$ $33$ $( 1,31,12, 4,29,10, 2,32,11, 3,30, 9)( 5,21,13, 8,24,15, 6,22,14, 7,23,16)(17,33,26,19,35,27,18,34,25,20,36,28)$
12C5 $12^{3}$ $2$ $12$ $33$ $( 1,32,12, 3,29, 9, 2,31,11, 4,30,10)( 5,22,13, 7,24,16, 6,21,14, 8,23,15)(17,34,26,20,35,28,18,33,25,19,36,27)$
12C-5 $12^{3}$ $2$ $12$ $33$ $( 1,20,21, 4,17,24, 2,19,22, 3,18,23)( 5,12,28, 8,10,26, 6,11,27, 7, 9,25)(13,29,33,15,32,35,14,30,34,16,31,36)$
12D1 $12^{3}$ $3$ $12$ $33$ $( 1,27,15, 4,25,14, 2,28,16, 3,26,13)( 5,21,19,35,31,12, 6,22,20,36,32,11)( 7,23,17,33,30, 9, 8,24,18,34,29,10)$
12D-1 $12^{3}$ $3$ $12$ $33$ $( 1, 9,26,33,16,23, 2,10,25,34,15,24)( 3,11,28,36,14,22, 4,12,27,35,13,21)( 5,17,32, 7,20,29, 6,18,31, 8,19,30)$
12D5 $12^{3}$ $3$ $12$ $33$ $( 1,14,26, 4,16,27, 2,13,25, 3,15,28)( 5,12,32,35,20,21, 6,11,31,36,19,22)( 7, 9,29,33,18,23, 8,10,30,34,17,24)$
12D-5 $12^{3}$ $3$ $12$ $33$ $( 1,23,15,33,25, 9, 2,24,16,34,26,10)( 3,22,13,36,27,11, 4,21,14,35,28,12)( 5,29,19, 7,31,17, 6,30,20, 8,32,18)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

36 x 36 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed