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Properties

Label 36T38
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3:D_{12}$

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Show commands: Magma

magma: G := TransitiveGroup(36, 38);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $38$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3:D_{12}$
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:  (1,30,16,5,26,19)(2,29,15,6,25,20)(3,31,14,7,28,17)(4,32,13,8,27,18)(9,36,24,10,35,23)(11,33,21)(12,34,22), (1,7,33,3,5,36,2,8,34,4,6,35)(9,26,17,21,14,30,10,25,18,22,13,29)(11,28,19,23,15,32,12,27,20,24,16,31)
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
$24$:  $D_{12}$, $(C_6\times C_2):C_2$
$36$:  $S_3^2$

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: $D_{12}$, $(C_6\times C_2):C_2$

Degree 18: $S_3^2$

Low degree siblings

12T38 x 2, 24T74, 36T33

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 $2^{15},1^{6}$ $6$ $2$ $15$ $( 3, 4)( 5,34)( 6,33)( 7,35)( 8,36)( 9,17)(10,18)(11,20)(12,19)(13,14)(21,29)(22,30)(23,32)(24,31)(27,28)$
2C $2^{18}$ $18$ $2$ $18$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,22)(10,21)(11,23)(12,24)(13,29)(14,30)(15,31)(16,32)(17,25)(18,26)(19,28)(20,27)(33,36)(34,35)$
3A $3^{12}$ $2$ $3$ $24$ $( 1,16,26)( 2,15,25)( 3,14,28)( 4,13,27)( 5,19,30)( 6,20,29)( 7,17,31)( 8,18,32)( 9,24,35)(10,23,36)(11,21,33)(12,22,34)$
3B $3^{12}$ $2$ $3$ $24$ $( 1,34, 5)( 2,33, 6)( 3,35, 8)( 4,36, 7)( 9,18,14)(10,17,13)(11,20,15)(12,19,16)(21,29,25)(22,30,26)(23,31,27)(24,32,28)$
3C $3^{12}$ $4$ $3$ $24$ $( 1,12,30)( 2,11,29)( 3, 9,32)( 4,10,31)( 5,16,22)( 6,15,21)( 7,13,23)( 8,14,24)(17,27,36)(18,28,35)(19,26,34)(20,25,33)$
4A $4^{9}$ $6$ $4$ $27$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,22,10,21)(11,24,12,23)(13,25,14,26)(15,28,16,27)(17,29,18,30)(19,31,20,32)(33,35,34,36)$
6A $6^{6}$ $2$ $6$ $30$ $( 1, 6,34, 2, 5,33)( 3, 7,35, 4, 8,36)( 9,13,18,10,14,17)(11,16,20,12,15,19)(21,26,29,22,25,30)(23,28,31,24,27,32)$
6B $6^{6}$ $2$ $6$ $30$ $( 1,15,26, 2,16,25)( 3,13,28, 4,14,27)( 5,20,30, 6,19,29)( 7,18,31, 8,17,32)( 9,23,35,10,24,36)(11,22,33,12,21,34)$
6C $6^{6}$ $4$ $6$ $30$ $( 1,20,22, 2,19,21)( 3,17,24, 4,18,23)( 5,11,26, 6,12,25)( 7, 9,27, 8,10,28)(13,32,36,14,31,35)(15,30,33,16,29,34)$
6D1 $6^{5},3^{2}$ $6$ $6$ $29$ $( 1,16,26)( 2,15,25)( 3,13,28, 4,14,27)( 5,12,30,34,19,22)( 6,11,29,33,20,21)( 7, 9,31,35,17,24)( 8,10,32,36,18,23)$
6D-1 $6^{5},3^{2}$ $6$ $6$ $29$ $( 1,26,16)( 2,25,15)( 3,27,14, 4,28,13)( 5,22,19,34,30,12)( 6,21,20,33,29,11)( 7,24,17,35,31, 9)( 8,23,18,36,32,10)$
12A1 $12^{3}$ $6$ $12$ $33$ $( 1,36, 6, 3,34, 7, 2,35, 5, 4,33, 8)( 9,30,13,21,18,26,10,29,14,22,17,25)(11,32,16,23,20,28,12,31,15,24,19,27)$
12A5 $12^{3}$ $6$ $12$ $33$ $( 1, 7,33, 3, 5,36, 2, 8,34, 4, 6,35)( 9,26,17,21,14,30,10,25,18,22,13,29)(11,28,19,23,15,32,12,27,20,24,16,31)$

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:  72.23
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 4A 6A 6B 6C 6D1 6D-1 12A1 12A5
Size 1 1 6 18 2 2 4 6 2 2 4 6 6 6 6
2 P 1A 1A 1A 1A 3A 3B 3C 2A 3B 3A 3C 3A 3A 6A 6A
3 P 1A 2A 2B 2C 1A 1A 1A 4A 2A 2A 2A 2B 2B 4A 4A
Type
72.23.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.23.1b R 1 1 −1 −1 1 1 1 1 1 1 1 −1 −1 1 1
72.23.1c R 1 1 −1 1 1 1 1 −1 1 1 1 −1 −1 −1 −1
72.23.1d R 1 1 1 −1 1 1 1 −1 1 1 1 1 1 −1 −1
72.23.2a R 2 2 0 0 2 −1 −1 2 −1 2 −1 0 0 −1 −1
72.23.2b R 2 2 2 0 −1 2 −1 0 2 −1 −1 −1 −1 0 0
72.23.2c R 2 −2 0 0 2 2 2 0 −2 −2 −2 0 0 0 0
72.23.2d R 2 2 −2 0 −1 2 −1 0 2 −1 −1 1 1 0 0
72.23.2e R 2 2 0 0 2 −1 −1 −2 −1 2 −1 0 0 1 1
72.23.2f1 R 2 −2 0 0 2 −1 −1 0 1 −2 1 0 0 −ζ12−1−ζ12 ζ12−1+ζ12
72.23.2f2 R 2 −2 0 0 2 −1 −1 0 1 −2 1 0 0 ζ12−1+ζ12 −ζ12−1−ζ12
72.23.2g1 C 2 −2 0 0 −1 2 −1 0 −2 1 1 −1−2ζ3 1+2ζ3 0 0
72.23.2g2 C 2 −2 0 0 −1 2 −1 0 −2 1 1 1+2ζ3 −1−2ζ3 0 0
72.23.4a R 4 4 0 0 −2 −2 1 0 −2 −2 1 0 0 0 0
72.23.4b R 4 −4 0 0 −2 −2 1 0 2 2 −1 0 0 0 0

magma: CharacterTable(G);