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Properties

Label 36T13
Degree $36$
Order $36$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3^2$

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Show commands: Magma

magma: G := TransitiveGroup(36, 13);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $13$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3^2$
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:  (1,5,33,3,7,36)(2,6,34,4,8,35)(9,28,18,22,16,29)(10,27,17,21,15,30)(11,25,20,23,13,31)(12,26,19,24,14,32), (1,13)(2,14)(3,15)(4,16)(5,10)(6,9)(7,11)(8,12)(17,36)(18,35)(19,34)(20,33)(21,29)(22,30)(23,32)(24,31)(25,26)(27,28)
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 2
$12$:  $D_{6}$ x 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: $S_3$ x 2, $D_{6}$ x 4, $S_3^2$

Degree 9: $S_3^2$

Degree 12: $D_6$ x 2, $S_3^2$

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

Low degree siblings

6T9, 9T8, 12T16, 18T9, 18T11 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 $2^{18}$ $3$ $2$ $18$ $( 1,26)( 2,25)( 3,28)( 4,27)( 5,29)( 6,30)( 7,32)( 8,31)( 9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,35)(22,36)(23,34)(24,33)$
2B $2^{18}$ $3$ $2$ $18$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,11)(10,12)(13,18)(14,17)(15,19)(16,20)(21,24)(22,23)(25,29)(26,30)(27,32)(28,31)(33,35)(34,36)$
2C $2^{18}$ $9$ $2$ $18$ $( 1,30)( 2,29)( 3,31)( 4,32)( 5,25)( 6,26)( 7,27)( 8,28)( 9,10)(11,12)(13,19)(14,20)(15,18)(16,17)(21,33)(22,34)(23,36)(24,35)$
3A $3^{12}$ $2$ $3$ $24$ $( 1, 7,33)( 2, 8,34)( 3, 5,36)( 4, 6,35)( 9,16,18)(10,15,17)(11,13,20)(12,14,19)(21,27,30)(22,28,29)(23,25,31)(24,26,32)$
3B $3^{12}$ $2$ $3$ $24$ $( 1,28,14)( 2,27,13)( 3,26,16)( 4,25,15)( 5,32,18)( 6,31,17)( 7,29,19)( 8,30,20)( 9,36,24)(10,35,23)(11,34,21)(12,33,22)$
3C $3^{12}$ $4$ $3$ $24$ $( 1,12,29)( 2,11,30)( 3, 9,32)( 4,10,31)( 5,16,24)( 6,15,23)( 7,14,22)( 8,13,21)(17,25,35)(18,26,36)(19,28,33)(20,27,34)$
6A $6^{6}$ $6$ $6$ $30$ $( 1,32,33,26, 7,24)( 2,31,34,25, 8,23)( 3,29,36,28, 5,22)( 4,30,35,27, 6,21)( 9,14,18,12,16,19)(10,13,17,11,15,20)$
6B $6^{6}$ $6$ $6$ $30$ $( 1,10,28,35,14,23)( 2, 9,27,36,13,24)( 3,11,26,34,16,21)( 4,12,25,33,15,22)( 5,20,32, 8,18,30)( 6,19,31, 7,17,29)$

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:  36.10
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 6A 6B
Size 1 3 3 9 2 2 4 6 6
2 P 1A 1A 1A 1A 3A 3B 3C 3A 3B
3 P 1A 2A 2B 2C 1A 1A 1A 2A 2B
Type
36.10.1a R 1 1 1 1 1 1 1 1 1
36.10.1b R 1 −1 −1 1 1 1 1 −1 −1
36.10.1c R 1 −1 1 −1 1 1 1 −1 1
36.10.1d R 1 1 −1 −1 1 1 1 1 −1
36.10.2a R 2 0 2 0 2 −1 −1 0 −1
36.10.2b R 2 2 0 0 −1 2 −1 −1 0
36.10.2c R 2 −2 0 0 −1 2 −1 1 0
36.10.2d R 2 0 −2 0 2 −1 −1 0 1
36.10.4a R 4 0 0 0 −2 −2 1 0 0

magma: CharacterTable(G);