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Properties

Label 36T572
36T572 1 4 1->4 13 1->13 2 3 2->3 14 2->14 3->1 15 3->15 4->2 16 4->16 5 10 5->10 34 5->34 6 9 6->9 33 6->33 7 11 7->11 35 7->35 8 12 8->12 36 8->36 25 9->25 9->35 26 10->26 10->36 27 11->27 11->34 28 12->28 12->33 13->8 22 13->22 14->7 21 14->21 15->5 23 15->23 16->6 24 16->24 17 17->4 32 17->32 18 18->3 31 18->31 19 19->2 29 19->29 20 20->1 30 20->30 21->12 21->25 22->11 22->26 23->9 23->27 24->10 24->28 25->17 25->22 26->18 26->21 27->19 27->24 28->20 28->23 29->13 29->31 30->14 30->32 31->16 31->30 32->15 32->29 33->5 33->19 34->6 34->20 35->8 35->18 36->7 36->17
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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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 572);
 

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 $t$:  $572$
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:  $(1,13,8,12,33,19,2,14,7,11,34,20)(3,15,5,10,36,17,4,16,6,9,35,18)(21,25,22,26)(23,27,24,28)(29,31,30,32)$, $(1,4,2,3)(5,34,6,33)(7,35,8,36)(9,25,17,32,15,23)(10,26,18,31,16,24)(11,27,19,29,13,22)(12,28,20,30,14,21)$
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: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_3$

Degree 9: $C_3^2 : C_6$

Degree 12: 12T30

Degree 18: 18T21

Low degree siblings

36T574, 36T656

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

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

38 x 38 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed