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Properties

Label 36T34
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times D_6$

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Show commands: Magma

magma: G := TransitiveGroup(36, 34);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$ x 2
$8$:  $C_2^3$
$12$:  $D_{6}$ x 6
$24$:  $S_3 \times C_2^2$ x 2
$36$:  $S_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$ x 2

Degree 4: $C_2^2$

Degree 6: $D_{6}$ x 6, $S_3^2$

Degree 9: $S_3^2$

Degree 12: $S_3 \times C_2^2$ x 2, 12T37

Degree 18: $S_3^2$, 18T29 x 2

Low degree siblings

12T37 x 2, 18T29 x 4, 24T73, 36T34, 36T40 x 4

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

Malle's constant $a(G)$:     $1/16$

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:  72.46
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 6A 6B 6C 6D 6E 6F 6G
Size 1 1 3 3 3 3 9 9 2 2 4 2 2 4 6 6 6 6
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A 3B 3C 3A 3B 3C 3B 3A 3A 3B
3 P 1A 2A 2D 2C 2E 2B 2F 2G 1A 1A 1A 2A 2A 2A 2B 2C 2D 2E
Type
72.46.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.46.1b R 1 −1 −1 −1 1 1 1 −1 1 1 1 −1 −1 −1 −1 −1 1 1
72.46.1c R 1 −1 −1 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 −1 1
72.46.1d R 1 −1 1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 1 −1 1 −1
72.46.1e R 1 −1 1 1 −1 −1 1 −1 1 1 1 −1 −1 −1 1 1 −1 −1
72.46.1f R 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
72.46.1g R 1 1 −1 1 1 −1 −1 −1 1 1 1 1 1 1 −1 1 1 −1
72.46.1h R 1 1 1 −1 −1 1 −1 −1 1 1 1 1 1 1 1 −1 −1 1
72.46.2a R 2 2 0 2 2 0 0 0 2 −1 −1 −1 2 −1 0 −1 −1 0
72.46.2b R 2 2 2 0 0 2 0 0 −1 2 −1 2 −1 −1 −1 0 0 −1
72.46.2c R 2 −2 −2 0 0 2 0 0 −1 2 −1 −2 1 1 1 0 0 −1
72.46.2d R 2 −2 0 −2 2 0 0 0 2 −1 −1 1 −2 1 0 1 −1 0
72.46.2e R 2 −2 0 2 −2 0 0 0 2 −1 −1 1 −2 1 0 −1 1 0
72.46.2f R 2 −2 2 0 0 −2 0 0 −1 2 −1 −2 1 1 −1 0 0 1
72.46.2g R 2 2 −2 0 0 −2 0 0 −1 2 −1 2 −1 −1 1 0 0 1
72.46.2h R 2 2 0 −2 −2 0 0 0 2 −1 −1 −1 2 −1 0 1 1 0
72.46.4a R 4 4 0 0 0 0 0 0 −2 −2 1 −2 −2 1 0 0 0 0
72.46.4b R 4 −4 0 0 0 0 0 0 −2 −2 1 2 2 −1 0 0 0 0

magma: CharacterTable(G);