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Properties

Label 36T21
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times A_4$

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Show commands: Magma

magma: G := TransitiveGroup(36, 21);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $21$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times A_4$
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:  (1,13,27,3,16,25)(2,14,28,4,15,26)(5,9,31,33,19,21)(6,10,32,34,20,22)(7,12,29,36,17,23)(8,11,30,35,18,24), (1,34,7)(2,33,8)(3,36,6)(4,35,5)(9,17,15,10,18,16)(11,20,14,12,19,13)(21,29,28,22,30,27)(23,31,25,24,32,26)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$12$:  $A_4$
$18$:  $S_3\times C_3$
$24$:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$, $S_3$

Degree 4: None

Degree 6: $C_6$, $S_3$, $A_4$, $S_3\times C_3$, $A_4\times C_2$

Degree 9: $S_3\times C_3$

Degree 12: $A_4 \times C_2$

Degree 18: $S_3 \times C_3$, 18T31, 18T32

Low degree siblings

12T43, 18T31, 18T32, 24T78, 24T83, 36T50, 36T51

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

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:  72.44
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B1 3B-1 3C1 3C-1 6A 6B1 6B-1
Size 1 3 3 9 2 4 4 8 8 6 12 12
2 P 1A 1A 1A 1A 3A 3B-1 3B1 3C-1 3C1 3A 3B1 3B-1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 2A 2B 2B
Type
72.44.1a R 1 1 1 1 1 1 1 1 1 1 1 1
72.44.1b R 1 1 −1 −1 1 1 1 1 1 1 −1 −1
72.44.1c1 C 1 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1
72.44.1c2 C 1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3
72.44.1d1 C 1 1 −1 −1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 −ζ3 −ζ3−1
72.44.1d2 C 1 1 −1 −1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 −ζ3−1 −ζ3
72.44.2a R 2 2 0 0 −1 2 2 −1 −1 −1 0 0
72.44.2b1 C 2 2 0 0 −1 2ζ3−1 2ζ3 −ζ3−1 −ζ3 −1 0 0
72.44.2b2 C 2 2 0 0 −1 2ζ3 2ζ3−1 −ζ3 −ζ3−1 −1 0 0
72.44.3a R 3 −1 3 −1 3 0 0 0 0 −1 0 0
72.44.3b R 3 −1 −3 1 3 0 0 0 0 −1 0 0
72.44.6a R 6 −2 0 0 −3 0 0 0 0 1 0 0

magma: CharacterTable(G);