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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

$\card{(G/N)}$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 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 $2^{18}$ $3$ $2$ $18$ $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,12)(10,11)(13,17)(14,18)(15,19)(16,20)(21,23)(22,24)(25,29)(26,30)(27,32)(28,31)(33,36)(34,35)$
2C $2^{18}$ $3$ $2$ $18$ $( 1, 3)( 2, 4)( 5,34)( 6,33)( 7,35)( 8,36)( 9,19)(10,20)(11,18)(12,17)(13,16)(14,15)(21,32)(22,31)(23,30)(24,29)(25,27)(26,28)$
3A1 $3^{12}$ $1$ $3$ $24$ $( 1,15,27)( 2,16,28)( 3,14,25)( 4,13,26)( 5,19,32)( 6,20,31)( 7,17,30)( 8,18,29)( 9,21,34)(10,22,33)(11,24,36)(12,23,35)$
3A-1 $3^{12}$ $1$ $3$ $24$ $( 1,27,15)( 2,28,16)( 3,25,14)( 4,26,13)( 5,32,19)( 6,31,20)( 7,30,17)( 8,29,18)( 9,34,21)(10,33,22)(11,36,24)(12,35,23)$
3B $3^{12}$ $2$ $3$ $24$ $( 1, 7,33)( 2, 8,34)( 3, 6,35)( 4, 5,36)( 9,16,18)(10,15,17)(11,13,19)(12,14,20)(21,28,29)(22,27,30)(23,25,31)(24,26,32)$
3C1 $3^{12}$ $2$ $3$ $24$ $( 1,30,10)( 2,29, 9)( 3,31,12)( 4,32,11)( 5,24,13)( 6,23,14)( 7,22,15)( 8,21,16)(17,33,27)(18,34,28)(19,36,26)(20,35,25)$
3C-1 $3^{12}$ $2$ $3$ $24$ $( 1,17,22)( 2,18,21)( 3,20,23)( 4,19,24)( 5,11,26)( 6,12,25)( 7,10,27)( 8, 9,28)(13,32,36)(14,31,35)(15,30,33)(16,29,34)$
6A1 $6^{6}$ $1$ $6$ $30$ $( 1,16,27, 2,15,28)( 3,13,25, 4,14,26)( 5,20,32, 6,19,31)( 7,18,30, 8,17,29)( 9,22,34,10,21,33)(11,23,36,12,24,35)$
6A-1 $6^{6}$ $1$ $6$ $30$ $( 1,28,15, 2,27,16)( 3,26,14, 4,25,13)( 5,31,19, 6,32,20)( 7,29,17, 8,30,18)( 9,33,21,10,34,22)(11,35,24,12,36,23)$
6B $6^{6}$ $2$ $6$ $30$ $( 1,21,17, 2,22,18)( 3,24,20, 4,23,19)( 5,25,11, 6,26,12)( 7,28,10, 8,27, 9)(13,35,32,14,36,31)(15,34,30,16,33,29)$
6C1 $6^{6}$ $2$ $6$ $30$ $( 1,34, 7, 2,33, 8)( 3,36, 6, 4,35, 5)( 9,17,16,10,18,15)(11,20,13,12,19,14)(21,30,28,22,29,27)(23,32,25,24,31,26)$
6C-1 $6^{6}$ $2$ $6$ $30$ $( 1, 9,30, 2,10,29)( 3,11,31, 4,12,32)( 5,14,24, 6,13,23)( 7,16,22, 8,15,21)(17,28,33,18,27,34)(19,25,36,20,26,35)$
6D1 $6^{6}$ $3$ $6$ $30$ $( 1,32,15, 5,27,19)( 2,31,16, 6,28,20)( 3,29,14, 8,25,18)( 4,30,13, 7,26,17)( 9,35,21,12,34,23)(10,36,22,11,33,24)$
6D-1 $6^{6}$ $3$ $6$ $30$ $( 1,19,27, 5,15,32)( 2,20,28, 6,16,31)( 3,18,25, 8,14,29)( 4,17,26, 7,13,30)( 9,23,34,12,21,35)(10,24,33,11,22,36)$
6E1 $6^{6}$ $3$ $6$ $30$ $( 1,25,15, 3,27,14)( 2,26,16, 4,28,13)( 5,21,19,34,32, 9)( 6,22,20,33,31,10)( 7,23,17,35,30,12)( 8,24,18,36,29,11)$
6E-1 $6^{6}$ $3$ $6$ $30$ $( 1,14,27, 3,15,25)( 2,13,28, 4,16,26)( 5, 9,32,34,19,21)( 6,10,31,33,20,22)( 7,12,30,35,17,23)( 8,11,29,36,18,24)$

Malle's constant $a(G)$:     $1/18$

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);