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Properties

Label 36T40
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times D_6$

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Show commands: Magma

magma: G := TransitiveGroup(36, 40);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $40$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times D_6$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $12$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,35,6,4,33,8)(2,36,5,3,34,7)(9,29,13,22,18,26)(10,30,14,21,17,25)(11,31,16,24,20,27)(12,32,15,23,19,28), (1,34)(2,33)(3,35)(4,36)(5,6)(7,8)(9,14)(10,13)(11,15)(12,16)(17,18)(19,20)(21,26)(22,25)(23,27)(24,28)(29,30)(31,32), (1,9)(2,10)(3,12)(4,11)(5,17)(6,18)(7,19)(8,20)(13,33)(14,34)(15,36)(16,35)(21,28)(22,27)(23,25)(24,26)(29,31)(30,32)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$ x 2
$8$:  $C_2^3$
$12$:  $D_{6}$ x 6
$24$:  $S_3 \times C_2^2$ x 2
$36$:  $S_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$ x 2

Degree 4: $C_2^2$

Degree 6: $S_3$, $D_{6}$ x 5

Degree 9: $S_3^2$

Degree 12: $D_6$, $S_3 \times C_2^2$

Degree 18: $S_3^2$, 18T29 x 2

Low degree siblings

12T37 x 2, 18T29 x 4, 24T73, 36T34 x 2, 36T40 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 $2^{18}$ $3$ $2$ $18$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,22)(10,21)(11,24)(12,23)(13,26)(14,25)(15,28)(16,27)(17,30)(18,29)(19,32)(20,31)(33,35)(34,36)$
2C $2^{18}$ $3$ $2$ $18$ $( 1,28)( 2,27)( 3,26)( 4,25)( 5,31)( 6,32)( 7,29)( 8,30)( 9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,35)(22,36)(23,33)(24,34)$
2D $2^{12},1^{12}$ $3$ $2$ $12$ $( 1,33)( 2,34)( 3,36)( 4,35)( 9,13)(10,14)(11,16)(12,15)(21,25)(22,26)(23,28)(24,27)$
2E $2^{18}$ $3$ $2$ $18$ $( 1,34)( 2,33)( 3,35)( 4,36)( 5, 6)( 7, 8)( 9,14)(10,13)(11,15)(12,16)(17,18)(19,20)(21,26)(22,25)(23,27)(24,28)(29,30)(31,32)$
2F $2^{18}$ $9$ $2$ $18$ $( 1,27)( 2,28)( 3,25)( 4,26)( 5,23)( 6,24)( 7,21)( 8,22)( 9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,35)(30,36)(31,33)(32,34)$
2G $2^{18}$ $9$ $2$ $18$ $( 1, 3)( 2, 4)( 5,35)( 6,36)( 7,33)( 8,34)( 9,30)(10,29)(11,32)(12,31)(13,25)(14,26)(15,27)(16,28)(17,22)(18,21)(19,24)(20,23)$
3A $3^{12}$ $2$ $3$ $24$ $( 1, 6,33)( 2, 5,34)( 3, 7,36)( 4, 8,35)( 9,13,18)(10,14,17)(11,16,20)(12,15,19)(21,25,30)(22,26,29)(23,28,32)(24,27,31)$
3B $3^{12}$ $2$ $3$ $24$ $( 1,26,16)( 2,25,15)( 3,28,14)( 4,27,13)( 5,30,19)( 6,29,20)( 7,32,17)( 8,31,18)( 9,35,24)(10,36,23)(11,33,22)(12,34,21)$
3C $3^{12}$ $4$ $3$ $24$ $( 1,29,11)( 2,30,12)( 3,32,10)( 4,31, 9)( 5,21,15)( 6,22,16)( 7,23,14)( 8,24,13)(17,36,28)(18,35,27)(19,34,25)(20,33,26)$
6A $6^{6}$ $2$ $6$ $30$ $( 1, 5,33, 2, 6,34)( 3, 8,36, 4, 7,35)( 9,14,18,10,13,17)(11,15,20,12,16,19)(21,26,30,22,25,29)(23,27,32,24,28,31)$
6B $6^{6}$ $2$ $6$ $30$ $( 1,15,26, 2,16,25)( 3,13,28, 4,14,27)( 5,20,30, 6,19,29)( 7,18,32, 8,17,31)( 9,23,35,10,24,36)(11,21,33,12,22,34)$
6C $6^{6}$ $4$ $6$ $30$ $( 1,19,22, 2,20,21)( 3,18,23, 4,17,24)( 5,11,25, 6,12,26)( 7, 9,28, 8,10,27)(13,32,35,14,31,36)(15,29,34,16,30,33)$
6D $6^{6}$ $6$ $6$ $30$ $( 1,21,16,34,26,12)( 2,22,15,33,25,11)( 3,24,14,35,28, 9)( 4,23,13,36,27,10)( 5,29,19, 6,30,20)( 7,31,17, 8,32,18)$
6E $6^{6}$ $6$ $6$ $30$ $( 1,23, 6,28,33,32)( 2,24, 5,27,34,31)( 3,22, 7,26,36,29)( 4,21, 8,25,35,30)( 9,19,13,12,18,15)(10,20,14,11,17,16)$
6F $6^{6}$ $6$ $6$ $30$ $( 1,35, 6, 4,33, 8)( 2,36, 5, 3,34, 7)( 9,29,13,22,18,26)(10,30,14,21,17,25)(11,31,16,24,20,27)(12,32,15,23,19,28)$
6G $6^{4},3^{4}$ $6$ $6$ $28$ $( 1,11,26,33,16,22)( 2,12,25,34,15,21)( 3,10,28,36,14,23)( 4, 9,27,35,13,24)( 5,19,30)( 6,20,29)( 7,17,32)( 8,18,31)$

Malle's constant $a(G)$:     $1/12$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  72.46
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 6A 6B 6C 6D 6E 6F 6G
Size 1 1 3 3 3 3 9 9 2 2 4 2 2 4 6 6 6 6
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A 3B 3C 3A 3B 3C 3B 3A 3A 3B
3 P 1A 2A 2D 2C 2E 2B 2F 2G 1A 1A 1A 2A 2A 2A 2B 2C 2D 2E
Type
72.46.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.46.1b R 1 −1 −1 −1 1 1 1 −1 1 1 1 −1 −1 −1 −1 −1 1 1
72.46.1c R 1 −1 −1 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 −1 1
72.46.1d R 1 −1 1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 1 −1 1 −1
72.46.1e R 1 −1 1 1 −1 −1 1 −1 1 1 1 −1 −1 −1 1 1 −1 −1
72.46.1f R 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
72.46.1g R 1 1 −1 1 1 −1 −1 −1 1 1 1 1 1 1 −1 1 1 −1
72.46.1h R 1 1 1 −1 −1 1 −1 −1 1 1 1 1 1 1 1 −1 −1 1
72.46.2a R 2 2 0 2 2 0 0 0 2 −1 −1 −1 2 −1 0 −1 −1 0
72.46.2b R 2 2 2 0 0 2 0 0 −1 2 −1 2 −1 −1 −1 0 0 −1
72.46.2c R 2 −2 −2 0 0 2 0 0 −1 2 −1 −2 1 1 1 0 0 −1
72.46.2d R 2 −2 0 −2 2 0 0 0 2 −1 −1 1 −2 1 0 1 −1 0
72.46.2e R 2 −2 0 2 −2 0 0 0 2 −1 −1 1 −2 1 0 −1 1 0
72.46.2f R 2 −2 2 0 0 −2 0 0 −1 2 −1 −2 1 1 −1 0 0 1
72.46.2g R 2 2 −2 0 0 −2 0 0 −1 2 −1 2 −1 −1 1 0 0 1
72.46.2h R 2 2 0 −2 −2 0 0 0 2 −1 −1 −1 2 −1 0 1 1 0
72.46.4a R 4 4 0 0 0 0 0 0 −2 −2 1 −2 −2 1 0 0 0 0
72.46.4b R 4 −4 0 0 0 0 0 0 −2 −2 1 2 2 −1 0 0 0 0

magma: CharacterTable(G);