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Properties

Label 36T112
36T112 1 8 1->8 9 1->9 14 1->14 2 7 2->7 10 2->10 13 2->13 3 5 3->5 12 3->12 16 3->16 4 6 4->6 11 4->11 15 4->15 17 5->17 18 5->18 6->17 6->18 19 7->19 20 7->20 8->19 8->20 9->10 22 9->22 26 9->26 21 10->21 25 10->25 24 11->24 28 11->28 23 12->23 27 12->27 13->20 13->21 13->25 14->19 14->22 14->26 15->18 15->24 15->28 16->17 16->23 16->27 31 17->31 32 17->32 18->31 18->32 29 19->29 30 19->30 20->29 20->30 21->4 34 21->34 22->3 33 22->33 23->2 23->24 35 23->35 24->1 36 24->36 25->4 25->29 25->34 26->3 26->30 26->33 27->2 27->32 27->35 28->1 28->31 28->36 29->5 29->6 30->5 30->6 31->7 31->8 32->7 32->8 33->12 33->16 33->34 34->11 34->15 35->10 35->13 36->9 36->14
Degree $36$
Order $144$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{12}:D_6$

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Copy content magma:G := TransitiveGroup(36, 112);
 

Group invariants

Abstract group:  $C_{12}:D_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $144=2^{4} \cdot 3^{2}$
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 $t$:  $112$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $6$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,14,26,3,16,27,2,13,25,4,15,28)(5,18,32,8,20,30,6,17,31,7,19,29)(9,22,33,12,23,35,10,21,34,11,24,36)$, $(1,9,26,33,16,23,2,10,25,34,15,24)(3,12,27,35,13,21,4,11,28,36,14,22)(5,17,32,7,20,29,6,18,31,8,19,30)$, $(1,8)(2,7)(3,5)(4,6)(9,10)(13,20)(14,19)(15,18)(16,17)(23,24)(25,29)(26,30)(27,32)(28,31)(33,34)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$3$:  $C_3$
$4$:  $C_2^2$ x 7
$6$:  $S_3$, $C_6$ x 7
$8$:  $D_{4}$ x 2, $C_2^3$
$12$:  $D_{6}$ x 3, $C_6\times C_2$ x 7
$16$:  $D_4\times C_2$
$18$:  $S_3\times C_3$
$24$:  $S_3 \times C_2^2$, $D_4 \times C_3$ x 2, 24T3
$36$:  $C_6\times S_3$ x 3
$48$:  12T28, 24T38
$72$:  24T68

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$, $S_3$

Degree 4: $D_{4}$

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

Degree 9: $S_3\times C_3$

Degree 12: $D_4 \times C_3$, 12T28

Degree 18: $S_3 \times C_6$

Low degree siblings

24T208 x 2, 36T112 x 3

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 $2^{18}$ $2$ $2$ $18$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)$
2C $2^{9},1^{18}$ $2$ $2$ $9$ $( 3, 4)( 5, 6)( 9,10)(13,14)(19,20)(23,24)(27,28)(31,32)(33,34)$
2D $2^{12},1^{12}$ $3$ $2$ $12$ $( 5,33)( 6,34)( 7,36)( 8,35)( 9,19)(10,20)(11,17)(12,18)(21,30)(22,29)(23,32)(24,31)$
2E $2^{18}$ $3$ $2$ $18$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,10)(11,12)(13,20)(14,19)(15,17)(16,18)(21,22)(23,24)(25,30)(26,29)(27,32)(28,31)(33,34)(35,36)$
2F $2^{18}$ $6$ $2$ $18$ $( 1, 4)( 2, 3)( 5,35)( 6,36)( 7,34)( 8,33)( 9,18)(10,17)(11,20)(12,19)(13,15)(14,16)(21,32)(22,31)(23,30)(24,29)(25,27)(26,28)$
2G $2^{15},1^{6}$ $6$ $2$ $15$ $( 1,36)( 2,35)( 3,34)( 4,33)( 7, 8)( 9,13)(10,14)(11,15)(12,16)(17,18)(21,25)(22,26)(23,28)(24,27)(29,30)$
3A1 $3^{12}$ $1$ $3$ $24$ $( 1,16,25)( 2,15,26)( 3,13,28)( 4,14,27)( 5,20,31)( 6,19,32)( 7,18,30)( 8,17,29)( 9,23,34)(10,24,33)(11,22,35)(12,21,36)$
3A-1 $3^{12}$ $1$ $3$ $24$ $( 1,25,16)( 2,26,15)( 3,28,13)( 4,27,14)( 5,31,20)( 6,32,19)( 7,30,18)( 8,29,17)( 9,34,23)(10,33,24)(11,35,22)(12,36,21)$
3B $3^{12}$ $2$ $3$ $24$ $( 1, 8,35)( 2, 7,36)( 3, 6,34)( 4, 5,33)( 9,13,19)(10,14,20)(11,16,17)(12,15,18)(21,26,30)(22,25,29)(23,28,32)(24,27,31)$
3C1 $3^{12}$ $2$ $3$ $24$ $( 1,17,22)( 2,18,21)( 3,19,23)( 4,20,24)( 5,10,27)( 6, 9,28)( 7,12,26)( 8,11,25)(13,32,34)(14,31,33)(15,30,36)(16,29,35)$
3C-1 $3^{12}$ $2$ $3$ $24$ $( 1,22,17)( 2,21,18)( 3,23,19)( 4,24,20)( 5,27,10)( 6,28, 9)( 7,26,12)( 8,25,11)(13,34,32)(14,33,31)(15,36,30)(16,35,29)$
4A $4^{9}$ $2$ $4$ $27$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)(17,20,18,19)(21,23,22,24)(25,27,26,28)(29,31,30,32)(33,36,34,35)$
4B $4^{9}$ $6$ $4$ $27$ $( 1,34, 2,33)( 3,36, 4,35)( 5, 8, 6, 7)( 9,15,10,16)(11,13,12,14)(17,19,18,20)(21,27,22,28)(23,26,24,25)(29,32,30,31)$
6A1 $6^{6}$ $1$ $6$ $30$ $( 1,26,16, 2,25,15)( 3,27,13, 4,28,14)( 5,32,20, 6,31,19)( 7,29,18, 8,30,17)( 9,33,23,10,34,24)(11,36,22,12,35,21)$
6A-1 $6^{6}$ $1$ $6$ $30$ $( 1,15,25, 2,16,26)( 3,14,28, 4,13,27)( 5,19,31, 6,20,32)( 7,17,30, 8,18,29)( 9,24,34,10,23,33)(11,21,35,12,22,36)$
6B $6^{6}$ $2$ $6$ $30$ $( 1, 7,35, 2, 8,36)( 3, 5,34, 4, 6,33)( 9,14,19,10,13,20)(11,15,17,12,16,18)(21,25,30,22,26,29)(23,27,32,24,28,31)$
6C1 $6^{6}$ $2$ $6$ $30$ $( 1,27,16, 4,25,14)( 2,28,15, 3,26,13)( 5,29,20, 8,31,17)( 6,30,19, 7,32,18)( 9,36,23,12,34,21)(10,35,24,11,33,22)$
6C-1 $6^{6}$ $2$ $6$ $30$ $( 1,14,25, 4,16,27)( 2,13,26, 3,15,28)( 5,17,31, 8,20,29)( 6,18,32, 7,19,30)( 9,21,34,12,23,36)(10,22,33,11,24,35)$
6D1 $6^{3},3^{6}$ $2$ $6$ $27$ $( 1,25,16)( 2,26,15)( 3,27,13, 4,28,14)( 5,32,20, 6,31,19)( 7,30,18)( 8,29,17)( 9,33,23,10,34,24)(11,35,22)(12,36,21)$
6D-1 $6^{3},3^{6}$ $2$ $6$ $27$ $( 1,16,25)( 2,15,26)( 3,14,28, 4,13,27)( 5,19,31, 6,20,32)( 7,18,30)( 8,17,29)( 9,24,34,10,23,33)(11,22,35)(12,21,36)$
6E1 $6^{6}$ $2$ $6$ $30$ $( 1,30,11, 2,29,12)( 3,31, 9, 4,32,10)( 5,23,14, 6,24,13)( 7,22,15, 8,21,16)(17,36,25,18,35,26)(19,33,28,20,34,27)$
6E-1 $6^{6}$ $2$ $6$ $30$ $( 1,12,29, 2,11,30)( 3,10,32, 4, 9,31)( 5,13,24, 6,14,23)( 7,16,21, 8,15,22)(17,26,35,18,25,36)(19,27,34,20,28,33)$
6F1 $6^{4},3^{4}$ $3$ $6$ $28$ $( 1,16,25)( 2,15,26)( 3,13,28)( 4,14,27)( 5,10,31,33,20,24)( 6, 9,32,34,19,23)( 7,12,30,36,18,21)( 8,11,29,35,17,22)$
6F-1 $6^{4},3^{4}$ $3$ $6$ $28$ $( 1,25,16)( 2,26,15)( 3,28,13)( 4,27,14)( 5,24,20,33,31,10)( 6,23,19,34,32, 9)( 7,21,18,36,30,12)( 8,22,17,35,29,11)$
6G1 $6^{6}$ $3$ $6$ $30$ $( 1,18,25, 7,16,30)( 2,17,26, 8,15,29)( 3,20,28, 5,13,31)( 4,19,27, 6,14,32)( 9,24,34,10,23,33)(11,21,35,12,22,36)$
6G-1 $6^{6}$ $3$ $6$ $30$ $( 1,30,16, 7,25,18)( 2,29,15, 8,26,17)( 3,31,13, 5,28,20)( 4,32,14, 6,27,19)( 9,33,23,10,34,24)(11,36,22,12,35,21)$
6H $6^{6}$ $4$ $6$ $30$ $( 1,33, 8, 4,35, 5)( 2,34, 7, 3,36, 6)( 9,18,13,12,19,15)(10,17,14,11,20,16)(21,32,26,23,30,28)(22,31,25,24,29,27)$
6I $6^{3},3^{6}$ $4$ $6$ $27$ $( 1,36, 8, 2,35, 7)( 3,34, 6)( 4,33, 5)( 9,19,13)(10,20,14)(11,18,16,12,17,15)(21,29,26,22,30,25)(23,32,28)(24,31,27)$
6J1 $6^{6}$ $4$ $6$ $30$ $( 1,24,17, 4,22,20)( 2,23,18, 3,21,19)( 5,25,10, 8,27,11)( 6,26, 9, 7,28,12)(13,36,32,15,34,30)(14,35,31,16,33,29)$
6J-1 $6^{6}$ $4$ $6$ $30$ $( 1,10,29, 4,11,31)( 2, 9,30, 3,12,32)( 5,16,24, 8,14,22)( 6,15,23, 7,13,21)(17,27,35,20,25,33)(18,28,36,19,26,34)$
6K1 $6^{3},3^{6}$ $4$ $6$ $27$ $( 1,21,17, 2,22,18)( 3,23,19)( 4,24,20)( 5,27,10)( 6,28, 9)( 7,25,12, 8,26,11)(13,34,32)(14,33,31)(15,35,30,16,36,29)$
6K-1 $6^{3},3^{6}$ $4$ $6$ $27$ $( 1,12,29, 2,11,30)( 3, 9,32)( 4,10,31)( 5,14,24)( 6,13,23)( 7,16,21, 8,15,22)(17,26,35,18,25,36)(19,28,34)(20,27,33)$
6L1 $6^{6}$ $6$ $6$ $30$ $( 1,27,16, 4,25,14)( 2,28,15, 3,26,13)( 5,22,20,35,31,11)( 6,21,19,36,32,12)( 7,23,18,34,30, 9)( 8,24,17,33,29,10)$
6L-1 $6^{6}$ $6$ $6$ $30$ $( 1,14,25, 4,16,27)( 2,13,26, 3,15,28)( 5,11,31,35,20,22)( 6,12,32,36,19,21)( 7, 9,30,34,18,23)( 8,10,29,33,17,24)$
6M1 $6^{5},3^{2}$ $6$ $6$ $29$ $( 1,21,16,36,25,12)( 2,22,15,35,26,11)( 3,23,13,34,28, 9)( 4,24,14,33,27,10)( 5,31,20)( 6,32,19)( 7,29,18, 8,30,17)$
6M-1 $6^{5},3^{2}$ $6$ $6$ $29$ $( 1,12,25,36,16,21)( 2,11,26,35,15,22)( 3, 9,28,34,13,23)( 4,10,27,33,14,24)( 5,20,31)( 6,19,32)( 7,17,30, 8,18,29)$
12A1 $12^{3}$ $2$ $12$ $33$ $( 1,13,26, 4,16,28, 2,14,25, 3,15,27)( 5,17,32, 7,20,29, 6,18,31, 8,19,30)( 9,21,33,11,23,36,10,22,34,12,24,35)$
12A-1 $12^{3}$ $2$ $12$ $33$ $( 1,28,15, 4,25,13, 2,27,16, 3,26,14)( 5,29,19, 7,31,17, 6,30,20, 8,32,18)( 9,36,24,11,34,21,10,35,23,12,33,22)$
12B $12^{3}$ $4$ $12$ $33$ $( 1,33, 7, 3,35, 5, 2,34, 8, 4,36, 6)( 9,17,14,12,19,16,10,18,13,11,20,15)(21,32,25,24,30,28,22,31,26,23,29,27)$
12C1 $12^{3}$ $4$ $12$ $33$ $( 1,10,30, 3,11,31, 2, 9,29, 4,12,32)( 5,15,23, 8,14,21, 6,16,24, 7,13,22)(17,27,36,19,25,33,18,28,35,20,26,34)$
12C-1 $12^{3}$ $4$ $12$ $33$ $( 1,24,18, 3,22,20, 2,23,17, 4,21,19)( 5,26, 9, 8,27,12, 6,25,10, 7,28,11)(13,35,31,15,34,29,14,36,32,16,33,30)$
12D1 $12^{3}$ $6$ $12$ $33$ $( 1,10,26,34,16,24, 2, 9,25,33,15,23)( 3,11,27,36,13,22, 4,12,28,35,14,21)( 5,18,32, 8,20,30, 6,17,31, 7,19,29)$
12D-1 $12^{3}$ $6$ $12$ $33$ $( 1,24,15,34,25,10, 2,23,16,33,26, 9)( 3,22,14,36,28,11, 4,21,13,35,27,12)( 5,30,19, 8,31,18, 6,29,20, 7,32,17)$

Malle's constant $a(G)$:     $1/9$

Copy content magma:ConjugacyClasses(G);
 

Character table

45 x 45 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed