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Properties

Label 36T23
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3:S_4$

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Show commands: Magma

magma: G := TransitiveGroup(36, 23);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $23$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3:S_4$
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:  (1,30,10)(2,29,9)(3,31,12)(4,32,11)(5,23,14)(6,24,13)(7,21,15)(8,22,16)(17,33,28)(18,34,27)(19,35,25)(20,36,26), (1,21,18)(2,22,17)(3,24,19)(4,23,20)(5,26,12)(6,25,11)(7,28,9)(8,27,10)(13,36,32)(14,35,31)(15,33,30)(16,34,29), (1,3)(2,4)(5,34)(6,33)(7,35)(8,36)(9,32)(10,31)(11,29)(12,30)(13,28)(14,27)(15,25)(16,26)(17,24)(18,23)(19,21)(20,22)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$ x 4
$18$:  $C_3^2:C_2$
$24$:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 4

Degree 4: None

Degree 6: $S_3$ x 4, $S_4$, $S_4$

Degree 9: $C_3^2:C_2$

Degree 12: $S_4$

Degree 18: $C_3^2 : C_2$, 18T37, 18T40

Low degree siblings

12T44 x 3, 18T37, 18T40, 24T79 x 3, 36T56

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

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:  72.43
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A 3B 3C 3D 4A 6A
Size 1 3 18 2 8 8 8 18 6
2 P 1A 1A 1A 3A 3B 3C 3D 2A 3A
3 P 1A 2A 2B 1A 1A 1A 1A 4A 2A
Type
72.43.1a R 1 1 1 1 1 1 1 1 1
72.43.1b R 1 1 −1 1 1 1 1 −1 1
72.43.2a R 2 2 0 −1 −1 −1 2 0 −1
72.43.2b R 2 2 0 −1 −1 2 −1 0 −1
72.43.2c R 2 2 0 −1 2 −1 −1 0 −1
72.43.2d R 2 2 0 2 −1 −1 −1 0 2
72.43.3a R 3 −1 1 3 0 0 0 −1 −1
72.43.3b R 3 −1 −1 3 0 0 0 1 −1
72.43.6a R 6 −2 0 −3 0 0 0 0 1

magma: CharacterTable(G);