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Properties

Label 36T39
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_6:S_3$

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Show commands: Magma

magma: G := TransitiveGroup(36, 39);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $39$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_6:S_3$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,8,34,4,6,35)(2,7,33,3,5,36)(9,26,18,22,14,29)(10,25,17,21,13,30)(11,27,20,23,15,31)(12,28,19,24,16,32), (1,24,2,23)(3,21,4,22)(5,31,6,32)(7,30,8,29)(9,15,10,16)(11,13,12,14)(17,19,18,20)(25,35,26,36)(27,34,28,33)
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
$8$:  $D_{4}$
$12$:  $D_{6}$ x 2
$24$:  $(C_6\times C_2):C_2$ x 2
$36$:  $S_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 2

Degree 4: $D_{4}$

Degree 6: $S_3$, $D_{6}$

Degree 9: $S_3^2$

Degree 12: $(C_6\times C_2):C_2$, $(C_6\times C_2):C_2$

Degree 18: $S_3^2$

Low degree siblings

24T61, 36T39

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

Malle's constant $a(G)$:     $1/15$

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:  72.22
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 4A 6A 6B 6C 6D1 6D-1 6E1 6E-1
Size 1 1 6 6 2 2 4 18 2 2 4 6 6 6 6
2 P 1A 1A 1A 1A 3A 3B 3C 2A 3A 3B 3C 3A 3B 3A 3B
3 P 1A 2A 2B 2C 1A 1A 1A 4A 2A 2A 2A 2B 2C 2B 2C
Type
72.22.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.22.1b R 1 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1
72.22.1c R 1 1 −1 1 1 1 1 −1 1 1 1 −1 −1 1 1
72.22.1d R 1 1 1 −1 1 1 1 −1 1 1 1 1 1 −1 −1
72.22.2a R 2 2 0 2 2 −1 −1 0 −1 2 −1 0 0 −1 −1
72.22.2b R 2 2 2 0 −1 2 −1 0 2 −1 −1 −1 −1 0 0
72.22.2c R 2 −2 0 0 2 2 2 0 −2 −2 −2 0 0 0 0
72.22.2d R 2 2 −2 0 −1 2 −1 0 2 −1 −1 1 1 0 0
72.22.2e R 2 2 0 −2 2 −1 −1 0 −1 2 −1 0 0 1 1
72.22.2f1 C 2 −2 0 0 −1 2 −1 0 −2 1 1 −1−2ζ3 1+2ζ3 0 0
72.22.2f2 C 2 −2 0 0 −1 2 −1 0 −2 1 1 1+2ζ3 −1−2ζ3 0 0
72.22.2g1 C 2 −2 0 0 2 −1 −1 0 1 −2 1 0 0 −1−2ζ3 1+2ζ3
72.22.2g2 C 2 −2 0 0 2 −1 −1 0 1 −2 1 0 0 1+2ζ3 −1−2ζ3
72.22.4a R 4 4 0 0 −2 −2 1 0 −2 −2 1 0 0 0 0
72.22.4b S 4 −4 0 0 −2 −2 1 0 2 2 −1 0 0 0 0

magma: CharacterTable(G);