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Properties

Label 36T12
Degree $36$
Order $36$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times A_4$

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Show commands: Magma

magma: G := TransitiveGroup(36, 12);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $12$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times A_4$
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:  (1,33,15)(2,34,16)(3,35,13)(4,36,14)(5,28,19)(6,27,20)(7,26,18)(8,25,17)(9,29,21)(10,30,22)(11,32,23)(12,31,24), (1,28,23)(2,27,24)(3,26,21)(4,25,22)(5,29,14)(6,30,13)(7,32,16)(8,31,15)(9,34,17)(10,33,18)(11,36,20)(12,35,19)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$ x 4
$9$:  $C_3^2$
$12$:  $A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$ x 4

Degree 4: $A_4$

Degree 6: $A_4$

Degree 9: $C_3^2$

Degree 12: $A_4$, $C_3\times A_4$ x 3

Degree 18: $A_4 \times C_3$

Low degree siblings

12T20 x 3, 18T8

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 $2^{18}$ $3$ $2$ $18$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,27)(26,28)(29,32)(30,31)(33,35)(34,36)$
3A1 $3^{12}$ $1$ $3$ $24$ $( 1, 6, 9)( 2, 5,10)( 3, 8,11)( 4, 7,12)(13,17,23)(14,18,24)(15,20,21)(16,19,22)(25,32,35)(26,31,36)(27,29,33)(28,30,34)$
3A-1 $3^{12}$ $1$ $3$ $24$ $( 1, 9, 6)( 2,10, 5)( 3,11, 8)( 4,12, 7)(13,23,17)(14,24,18)(15,21,20)(16,22,19)(25,35,32)(26,36,31)(27,33,29)(28,34,30)$
3B1 $3^{12}$ $4$ $3$ $24$ $( 1,27,21)( 2,28,22)( 3,25,23)( 4,26,24)( 5,30,16)( 6,29,15)( 7,31,14)( 8,32,13)( 9,33,20)(10,34,19)(11,35,17)(12,36,18)$
3B-1 $3^{12}$ $4$ $3$ $24$ $( 1,21,27)( 2,22,28)( 3,23,25)( 4,24,26)( 5,16,30)( 6,15,29)( 7,14,31)( 8,13,32)( 9,20,33)(10,19,34)(11,17,35)(12,18,36)$
3C1 $3^{12}$ $4$ $3$ $24$ $( 1,29,20)( 2,30,19)( 3,32,17)( 4,31,18)( 5,34,22)( 6,33,21)( 7,36,24)( 8,35,23)( 9,27,15)(10,28,16)(11,25,13)(12,26,14)$
3C-1 $3^{12}$ $4$ $3$ $24$ $( 1,33,15)( 2,34,16)( 3,35,13)( 4,36,14)( 5,28,19)( 6,27,20)( 7,26,18)( 8,25,17)( 9,29,21)(10,30,22)(11,32,23)(12,31,24)$
3D1 $3^{12}$ $4$ $3$ $24$ $( 1,15,33)( 2,16,34)( 3,13,35)( 4,14,36)( 5,19,28)( 6,20,27)( 7,18,26)( 8,17,25)( 9,21,29)(10,22,30)(11,23,32)(12,24,31)$
3D-1 $3^{12}$ $4$ $3$ $24$ $( 1,20,29)( 2,19,30)( 3,17,32)( 4,18,31)( 5,22,34)( 6,21,33)( 7,24,36)( 8,23,35)( 9,15,27)(10,16,28)(11,13,25)(12,14,26)$
6A1 $6^{6}$ $3$ $6$ $30$ $( 1,12, 6, 4, 9, 7)( 2,11, 5, 3,10, 8)(13,24,17,14,23,18)(15,22,20,16,21,19)(25,33,32,27,35,29)(26,34,31,28,36,30)$
6A-1 $6^{6}$ $3$ $6$ $30$ $( 1, 7, 9, 4, 6,12)( 2, 8,10, 3, 5,11)(13,18,23,14,17,24)(15,19,21,16,20,22)(25,29,35,27,32,33)(26,30,36,28,31,34)$

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:  36.11
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 6A1 6A-1
Size 1 3 1 1 4 4 4 4 4 4 3 3
2 P 1A 1A 3A-1 3A1 3B-1 3B1 3C-1 3D-1 3D1 3C1 3A1 3A-1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A
Type
36.11.1a R 1 1 1 1 1 1 1 1 1 1 1 1
36.11.1b1 C 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 ζ3−1
36.11.1b2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 ζ3
36.11.1c1 C 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1
36.11.1c2 C 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
36.11.1d1 C 1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1
36.11.1d2 C 1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
36.11.1e1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1
36.11.1e2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1
36.11.3a R 3 −1 3 3 0 0 0 0 0 0 −1 −1
36.11.3b1 C 3 −1 3ζ3−1 3ζ3 0 0 0 0 0 0 −ζ3 −ζ3−1
36.11.3b2 C 3 −1 3ζ3 3ζ3−1 0 0 0 0 0 0 −ζ3−1 −ζ3

magma: CharacterTable(G);