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Properties

Label 36T134
Degree $36$
Order $144$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_6:S_4$

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Copy content magma:G := TransitiveGroup(36, 134);
 

Group invariants

Abstract group:  $C_6:S_4$
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 $t$:  $134$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,28,13,3,25,16)(2,27,14,4,26,15)(5,31,17,8,30,19)(6,32,18,7,29,20)(9,36,23,12,33,22)(10,35,24,11,34,21)$, $(1,33,8,3,36,5)(2,34,7,4,35,6)(9,20,16,12,18,13)(10,19,15,11,17,14)(21,30,25,24,31,28)(22,29,26,23,32,27)$, $(1,9)(2,10)(3,12)(4,11)(5,20)(6,19)(7,17)(8,18)(13,33)(14,34)(15,35)(16,36)(21,27)(22,28)(23,25)(24,26)(29,31)(30,32)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 4
$12$:  $D_{6}$ x 4
$18$:  $C_3^2:C_2$
$24$:  $S_4$
$36$:  18T12
$48$:  $S_4\times C_2$
$72$:  12T44

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 4

Degree 4: None

Degree 6: $D_{6}$ x 4, $S_4$, $S_4\times C_2$

Degree 9: $C_3^2:C_2$

Degree 12: $C_2 \times S_4$

Degree 18: 18T12, 18T40, 18T66

Low degree siblings

18T66 x 2, 24T251 x 6, 36T109, 36T133 x 2, 36T134, 36T166

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 $2^{18}$ $1$ $2$ $18$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)$
2B $2^{12},1^{12}$ $3$ $2$ $12$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(21,22)(23,24)(29,30)(31,32)$
2C $2^{18}$ $3$ $2$ $18$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32)(33,36)(34,35)$
2D $2^{18}$ $18$ $2$ $18$ $( 1,16)( 2,15)( 3,13)( 4,14)( 5,12)( 6,11)( 7,10)( 8, 9)(17,35)(18,36)(19,34)(20,33)(21,29)(22,30)(23,31)(24,32)(25,27)(26,28)$
2E $2^{18}$ $18$ $2$ $18$ $( 1, 2)( 3, 4)( 5,33)( 6,34)( 7,35)( 8,36)( 9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28)(17,23)(18,24)(19,22)(20,21)$
3A $3^{12}$ $2$ $3$ $24$ $( 1,32,11)( 2,31,12)( 3,29,10)( 4,30, 9)( 5,23,15)( 6,24,16)( 7,21,13)( 8,22,14)(17,33,27)(18,34,28)(19,36,26)(20,35,25)$
3B $3^{12}$ $8$ $3$ $24$ $( 1,13,25)( 2,14,26)( 3,16,28)( 4,15,27)( 5,17,30)( 6,18,29)( 7,20,32)( 8,19,31)( 9,23,33)(10,24,34)(11,21,35)(12,22,36)$
3C $3^{12}$ $8$ $3$ $24$ $( 1, 7,35)( 2, 8,36)( 3, 6,34)( 4, 5,33)( 9,15,17)(10,16,18)(11,13,20)(12,14,19)(21,25,32)(22,26,31)(23,27,30)(24,28,29)$
3D $3^{12}$ $8$ $3$ $24$ $( 1,21,20)( 2,22,19)( 3,24,18)( 4,23,17)( 5,27, 9)( 6,28,10)( 7,25,11)( 8,26,12)(13,35,32)(14,36,31)(15,33,30)(16,34,29)$
4A $4^{6},2^{4},1^{4}$ $18$ $4$ $22$ $( 1,36, 2,35)( 3,33, 4,34)( 9,28,10,27)(11,26,12,25)(13,21)(14,22)(15,23)(16,24)(17,30,18,29)(19,31,20,32)$
4B $4^{6},2^{6}$ $18$ $4$ $24$ $( 1,17, 2,18)( 3,19, 4,20)( 5,14)( 6,13)( 7,16)( 8,15)( 9,35,10,36)(11,33,12,34)(21,24)(22,23)(25,29,26,30)(27,31,28,32)$
6A $6^{6}$ $2$ $6$ $30$ $( 1,10,32, 3,11,29)( 2, 9,31, 4,12,30)( 5,14,23, 8,15,22)( 6,13,24, 7,16,21)(17,26,33,19,27,36)(18,25,34,20,28,35)$
6B $6^{4},3^{4}$ $6$ $6$ $28$ $( 1,12,32, 2,11,31)( 3, 9,29, 4,10,30)( 5,16,23, 6,15,24)( 7,14,21, 8,13,22)(17,27,33)(18,28,34)(19,26,36)(20,25,35)$
6C $6^{6}$ $6$ $6$ $30$ $( 1, 9,32, 4,11,30)( 2,10,31, 3,12,29)( 5,13,23, 7,15,21)( 6,14,24, 8,16,22)(17,26,33,19,27,36)(18,25,34,20,28,35)$
6D $6^{6}$ $8$ $6$ $30$ $( 1,28,13, 3,25,16)( 2,27,14, 4,26,15)( 5,31,17, 8,30,19)( 6,32,18, 7,29,20)( 9,36,23,12,33,22)(10,35,24,11,34,21)$
6E $6^{6}$ $8$ $6$ $30$ $( 1,34, 7, 3,35, 6)( 2,33, 8, 4,36, 5)( 9,19,15,12,17,14)(10,20,16,11,18,13)(21,29,25,24,32,28)(22,30,26,23,31,27)$
6F $6^{6}$ $8$ $6$ $30$ $( 1,18,21, 3,20,24)( 2,17,22, 4,19,23)( 5,12,27, 8, 9,26)( 6,11,28, 7,10,25)(13,29,35,16,32,34)(14,30,36,15,31,33)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 6E 6F
Size 1 1 3 3 18 18 2 8 8 8 18 18 2 6 6 8 8 8
2 P 1A 1A 1A 1A 1A 1A 3A 3B 3C 3D 2B 2B 3A 3A 3A 3B 3C 3D
3 P 1A 2A 2B 2C 2D 2E 1A 1A 1A 1A 4A 4B 2A 2B 2C 2A 2A 2A
Type
144.189.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
144.189.1b R 1 −1 1 −1 −1 1 1 1 1 1 1 −1 −1 1 −1 −1 −1 −1
144.189.1c R 1 −1 1 −1 1 −1 1 1 1 1 −1 1 −1 1 −1 −1 −1 −1
144.189.1d R 1 1 1 1 −1 −1 1 1 1 1 −1 −1 1 1 1 1 1 1
144.189.2a R 2 2 2 2 0 0 −1 −1 −1 2 0 0 −1 −1 −1 −1 −1 2
144.189.2b R 2 2 2 2 0 0 −1 −1 2 −1 0 0 −1 −1 −1 −1 2 −1
144.189.2c R 2 2 2 2 0 0 −1 2 −1 −1 0 0 −1 −1 −1 2 −1 −1
144.189.2d R 2 2 2 2 0 0 2 −1 −1 −1 0 0 2 2 2 −1 −1 −1
144.189.2e R 2 −2 2 −2 0 0 −1 −1 −1 2 0 0 1 −1 1 1 1 −2
144.189.2f R 2 −2 2 −2 0 0 −1 −1 2 −1 0 0 1 −1 1 1 −2 1
144.189.2g R 2 −2 2 −2 0 0 −1 2 −1 −1 0 0 1 −1 1 −2 1 1
144.189.2h R 2 −2 2 −2 0 0 2 −1 −1 −1 0 0 −2 2 −2 1 1 1
144.189.3a R 3 3 −1 −1 1 1 3 0 0 0 −1 −1 3 −1 −1 0 0 0
144.189.3b R 3 3 −1 −1 −1 −1 3 0 0 0 1 1 3 −1 −1 0 0 0
144.189.3c R 3 −3 −1 1 −1 1 3 0 0 0 −1 1 −3 −1 1 0 0 0
144.189.3d R 3 −3 −1 1 1 −1 3 0 0 0 1 −1 −3 −1 1 0 0 0
144.189.6a R 6 6 −2 −2 0 0 −3 0 0 0 0 0 −3 1 1 0 0 0
144.189.6b R 6 −6 −2 2 0 0 −3 0 0 0 0 0 3 1 −1 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed