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Properties

Label 36T6
Degree $36$
Order $36$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_6\times S_3$

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Show commands: Magma

magma: G := TransitiveGroup(36, 6);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $6$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_6\times S_3$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $36$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,29,10,2,30,9)(3,32,12,4,31,11)(5,23,13,6,24,14)(7,21,15,8,22,16)(17,34,27,18,33,28)(19,35,26,20,36,25), (1,25,15,3,27,14)(2,26,16,4,28,13)(5,21,19,34,32,9)(6,22,20,33,31,10)(7,23,17,35,30,12)(8,24,18,36,29,11)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $C_3$, $S_3$

Degree 4: $C_2^2$

Degree 6: $C_6$ x 3, $S_3$, $D_{6}$ x 2, $S_3\times C_3$

Degree 9: $S_3\times C_3$

Degree 12: $C_6\times C_2$, $D_6$, $C_6\times S_3$

Degree 18: $S_3 \times C_3$, $S_3 \times C_6$ x 2

Low degree siblings

12T18, 18T6 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  36.12
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 3B 3C1 3C-1 6A1 6A-1 6B 6C1 6C-1 6D1 6D-1 6E1 6E-1
Size 1 1 3 3 1 1 2 2 2 1 1 2 2 2 3 3 3 3
2 P 1A 1A 1A 1A 3A-1 3A1 3B 3C-1 3C1 3A-1 3A1 3C-1 3B 3C1 3A1 3A-1 3A1 3A-1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2B 2B 2C 2C
Type
36.12.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
36.12.1b R 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 1 1
36.12.1c R 1 −1 1 −1 1 1 1 1 1 −1 −1 −1 −1 −1 1 1 −1 −1
36.12.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
36.12.1e1 C 1 1 1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1
36.12.1e2 C 1 1 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3
36.12.1f1 C 1 −1 −1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3 ζ3−1
36.12.1f2 C 1 −1 −1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3−1 ζ3
36.12.1g1 C 1 −1 1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −1 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1
36.12.1g2 C 1 −1 1 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −1 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3
36.12.1h1 C 1 1 −1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1
36.12.1h2 C 1 1 −1 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3
36.12.2a R 2 2 0 0 2 2 −1 −1 −1 2 2 −1 −1 −1 0 0 0 0
36.12.2b R 2 −2 0 0 2 2 −1 −1 −1 −2 −2 1 1 1 0 0 0 0
36.12.2c1 C 2 2 0 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 2ζ3−1 2ζ3 −1 −ζ3 −ζ3−1 0 0 0 0
36.12.2c2 C 2 2 0 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 2ζ3 2ζ3−1 −1 −ζ3−1 −ζ3 0 0 0 0
36.12.2d1 C 2 −2 0 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 −2ζ3−1 −2ζ3 1 ζ3 ζ3−1 0 0 0 0
36.12.2d2 C 2 −2 0 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 −2ζ3 −2ζ3−1 1 ζ3−1 ζ3 0 0 0 0

magma: CharacterTable(G);