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Properties

Label 36T656
36T656 1 13 1->13 32 1->32 2 14 2->14 31 2->31 3 15 3->15 29 3->29 4 16 4->16 30 4->30 5 24 5->24 33 5->33 6 23 6->23 34 6->34 7 22 7->22 35 7->35 8 21 8->21 36 8->36 9 20 9->20 26 9->26 10 19 10->19 25 10->25 11 17 11->17 28 11->28 12 18 12->18 27 12->27 13->3 13->4 14->3 14->4 15->1 15->1 16->2 16->2 17->6 17->10 18->5 18->9 19->8 19->11 20->7 20->12 21->5 21->9 22->6 22->10 23->7 23->12 24->8 24->11 25->21 25->29 26->22 26->30 27->24 27->32 28->23 28->31 29->13 30->14 31->15 32->16 33->17 33->34 34->18 35->19 36->20
Degree $36$
Order $432$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $(C_3\times A_4):C_{12}$

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Copy content magma:G := TransitiveGroup(36, 656);
 

Group invariants

Abstract group:  $(C_3\times A_4):C_{12}$
Copy content magma:IdentifyGroup(G);
 
Order:  $432=2^{4} \cdot 3^{3}$
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 $t$:  $656$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,13,3,15)(2,14,4,16)(5,24,8,21)(6,23,7,22)(9,20,12,18)(10,19,11,17)(25,29)(26,30)(27,32)(28,31)(33,34)$, $(1,32,16,2,31,15)(3,29,13,4,30,14)(5,33,17,6,34,18)(7,35,19,8,36,20)(9,26,22,10,25,21)(11,28,23,12,27,24)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $S_3$, $C_6$
$12$:  $C_{12}$, $C_3 : C_4$
$18$:  $S_3\times C_3$
$24$:  $S_4$
$36$:  $C_3\times (C_3 : C_4)$
$48$:  12T27
$54$:  $C_3^2 : C_6$
$72$:  12T45
$108$:  36T70
$144$:  36T104
$216$:  18T97

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4$

Degree 9: $C_3^2 : C_6$

Degree 12: 12T27

Degree 18: 18T97

Low degree siblings

36T572, 36T574

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 $2^{18}$ $3$ $2$ $18$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)$
2C $2^{12},1^{12}$ $3$ $2$ $12$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23)$
3A $3^{12}$ $2$ $3$ $24$ $( 1,12, 7)( 2,11, 8)( 3, 9, 6)( 4,10, 5)(13,22,18)(14,21,17)(15,23,20)(16,24,19)(25,33,30)(26,34,29)(27,35,32)(28,36,31)$
3B1 $3^{8},1^{12}$ $3$ $3$ $16$ $(13,18,22)(14,17,21)(15,20,23)(16,19,24)(25,33,30)(26,34,29)(27,35,32)(28,36,31)$
3B-1 $3^{8},1^{12}$ $3$ $3$ $16$ $(13,22,18)(14,21,17)(15,23,20)(16,24,19)(25,30,33)(26,29,34)(27,32,35)(28,31,36)$
3C $3^{12}$ $24$ $3$ $24$ $( 1,27,21)( 2,28,22)( 3,26,23)( 4,25,24)( 5,30,16)( 6,29,15)( 7,32,14)( 8,31,13)( 9,34,20)(10,33,19)(11,36,18)(12,35,17)$
3D1 $3^{12}$ $24$ $3$ $24$ $( 1,17,27)( 2,18,28)( 3,20,26)( 4,19,25)( 5,24,30)( 6,23,29)( 7,21,32)( 8,22,31)( 9,15,34)(10,16,33)(11,13,36)(12,14,35)$
3D-1 $3^{12}$ $24$ $3$ $24$ $( 1,27,17)( 2,28,18)( 3,26,20)( 4,25,19)( 5,30,24)( 6,29,23)( 7,32,21)( 8,31,22)( 9,34,15)(10,33,16)(11,36,13)(12,35,14)$
4A1 $4^{9}$ $18$ $4$ $27$ $( 1, 4, 2, 3)( 5,11, 6,12)( 7,10, 8, 9)(13,34,14,33)(15,35,16,36)(17,30,18,29)(19,31,20,32)(21,25,22,26)(23,27,24,28)$
4A-1 $4^{9}$ $18$ $4$ $27$ $( 1, 3, 2, 4)( 5,12, 6,11)( 7, 9, 8,10)(13,33,14,34)(15,36,16,35)(17,29,18,30)(19,32,20,31)(21,26,22,25)(23,28,24,27)$
4B1 $4^{6},2^{5},1^{2}$ $18$ $4$ $23$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 7, 8)(13,31,16,29)(14,32,15,30)(17,27,20,25)(18,28,19,26)(21,35,23,33)(22,36,24,34)$
4B-1 $4^{6},2^{5},1^{2}$ $18$ $4$ $23$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 7, 8)(13,29,16,31)(14,30,15,32)(17,25,20,27)(18,26,19,28)(21,33,23,35)(22,34,24,36)$
6A $6^{6}$ $2$ $6$ $30$ $( 1, 8,12, 2, 7,11)( 3, 5, 9, 4, 6,10)(13,17,22,14,18,21)(15,19,23,16,20,24)(25,29,33,26,30,34)(27,31,35,28,32,36)$
6B1 $6^{4},2^{6}$ $3$ $6$ $26$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,23,18,15,22,20)(14,24,17,16,21,19)(25,31,33,28,30,36)(26,32,34,27,29,35)$
6B-1 $6^{4},2^{6}$ $3$ $6$ $26$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,20,22,15,18,23)(14,19,21,16,17,24)(25,36,30,28,33,31)(26,35,29,27,34,32)$
6C1 $6^{4},2^{6}$ $3$ $6$ $26$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,21,18,14,22,17)(15,24,20,16,23,19)(25,29,33,26,30,34)(27,31,35,28,32,36)$
6C-1 $6^{4},2^{6}$ $3$ $6$ $26$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,17,22,14,18,21)(15,19,23,16,20,24)(25,34,30,26,33,29)(27,36,32,28,35,31)$
6D1 $6^{4},1^{12}$ $3$ $6$ $20$ $(13,24,18,16,22,19)(14,23,17,15,21,20)(25,32,33,27,30,35)(26,31,34,28,29,36)$
6D-1 $6^{4},1^{12}$ $3$ $6$ $20$ $(13,19,22,16,18,24)(14,20,21,15,17,23)(25,35,30,27,33,32)(26,36,29,28,34,31)$
6E $6^{6}$ $6$ $6$ $30$ $( 1,10, 7, 4,12, 5)( 2, 9, 8, 3,11, 6)(13,24,18,16,22,19)(14,23,17,15,21,20)(25,34,30,26,33,29)(27,36,32,28,35,31)$
6F $6^{4},3^{4}$ $6$ $6$ $28$ $( 1, 9, 7, 3,12, 6)( 2,10, 8, 4,11, 5)(13,23,18,15,22,20)(14,24,17,16,21,19)(25,33,30)(26,34,29)(27,35,32)(28,36,31)$
6G1 $6^{4},2^{6}$ $6$ $6$ $26$ $( 1, 8,12, 2, 7,11)( 3, 5, 9, 4, 6,10)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,36,30,28,33,31)(26,35,29,27,34,32)$
6G-1 $6^{4},2^{6}$ $6$ $6$ $26$ $( 1,10, 7, 4,12, 5)( 2, 9, 8, 3,11, 6)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,29,33,26,30,34)(27,31,35,28,32,36)$
6H1 $6^{2},3^{4},2^{6}$ $6$ $6$ $24$ $( 1, 7,12)( 2, 8,11)( 3, 6, 9)( 4, 5,10)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,35,30,27,33,32)(26,36,29,28,34,31)$
6H-1 $6^{2},3^{4},2^{6}$ $6$ $6$ $24$ $( 1, 9, 7, 3,12, 6)( 2,10, 8, 4,11, 5)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,30,33)(26,29,34)(27,32,35)(28,31,36)$
6I $6^{6}$ $24$ $6$ $30$ $( 1,22,27, 2,21,28)( 3,24,26, 4,23,25)( 5,15,30, 6,16,29)( 7,13,32, 8,14,31)( 9,19,34,10,20,33)(11,17,36,12,18,35)$
6J1 $6^{6}$ $24$ $6$ $30$ $( 1,28,17, 2,27,18)( 3,25,20, 4,26,19)( 5,29,24, 6,30,23)( 7,31,21, 8,32,22)( 9,33,15,10,34,16)(11,35,13,12,36,14)$
6J-1 $6^{6}$ $24$ $6$ $30$ $( 1,18,27, 2,17,28)( 3,19,26, 4,20,25)( 5,23,30, 6,24,29)( 7,22,32, 8,21,31)( 9,16,34,10,15,33)(11,14,36,12,13,35)$
12A1 $12^{2},4^{3}$ $18$ $12$ $31$ $( 1, 3, 2, 4)( 5,12, 6,11)( 7, 9, 8,10)(13,30,21,34,18,25,14,29,22,33,17,26)(15,31,24,35,20,28,16,32,23,36,19,27)$
12A-1 $12^{2},4^{3}$ $18$ $12$ $31$ $( 1, 3, 2, 4)( 5,12, 6,11)( 7, 9, 8,10)(13,27,17,36,22,32,14,28,18,35,21,31)(15,26,19,33,23,29,16,25,20,34,24,30)$
12A5 $12^{2},4^{3}$ $18$ $12$ $31$ $( 1, 4, 2, 3)( 5,11, 6,12)( 7,10, 8, 9)(13,28,17,35,22,31,14,27,18,36,21,32)(15,25,19,34,23,30,16,26,20,33,24,29)$
12A-5 $12^{2},4^{3}$ $18$ $12$ $31$ $( 1, 4, 2, 3)( 5,11, 6,12)( 7,10, 8, 9)(13,29,21,33,18,26,14,30,22,34,17,25)(15,32,24,36,20,27,16,31,23,35,19,28)$
12B1 $12^{2},2^{5},1^{2}$ $18$ $12$ $27$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 7, 8)(13,26,24,31,18,34,16,28,22,29,19,36)(14,25,23,32,17,33,15,27,21,30,20,35)$
12B-1 $12^{2},2^{5},1^{2}$ $18$ $12$ $27$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 7, 8)(13,36,19,29,22,28,16,34,18,31,24,26)(14,35,20,30,21,27,15,33,17,32,23,25)$
12B5 $12^{2},2^{5},1^{2}$ $18$ $12$ $27$ $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 6)(13,35,19,30,22,27,16,33,18,32,24,25)(14,36,20,29,21,28,15,34,17,31,23,26)$
12B-5 $12^{2},2^{5},1^{2}$ $18$ $12$ $27$ $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 6)(13,25,24,32,18,33,16,27,22,30,19,35)(14,26,23,31,17,34,15,28,21,29,20,36)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

38 x 38 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed