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Properties

Label 36T19
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_6\wr C_2$

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Show commands: Magma

magma: G := TransitiveGroup(36, 19);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$8$:  $D_{4}$
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$24$:  $(C_6\times C_2):C_2$, $D_4 \times C_3$
$36$:  $C_6\times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$, $S_3$

Degree 4: $D_{4}$

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

Degree 9: $S_3\times C_3$

Degree 12: $D_4 \times C_3$, $(C_6\times C_2):C_2$, 12T42 x 2

Degree 18: $S_3 \times C_3$

Low degree siblings

12T42 x 2, 24T77, 36T26

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

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:  72.30
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 3B 3C1 3C-1 4A 6A1 6A-1 6B 6C1 6C-1 6D1 6D-1 6E1 6E-1 6F1 6F-1 6G1 6G-1 6H1 6H-1 12A1 12A-1
Size 1 1 2 6 1 1 2 2 2 6 1 1 2 2 2 2 2 2 2 2 2 2 2 6 6 6 6
2 P 1A 1A 1A 1A 3A-1 3A1 3B 3C-1 3C1 2A 3A1 3A-1 3C1 3C-1 3A-1 3B 3C-1 3A1 3C1 3C1 3B 3C-1 3B 3A-1 3A1 6A1 6A-1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 4A 2A 2A 2A 2B 2B 2B 2B 2B 2B 2B 2A 2A 2B 2C 2C 4A 4A
Type
72.30.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.30.1b R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 1 1
72.30.1c R 1 1 −1 1 1 1 1 1 1 −1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 −1 −1
72.30.1d R 1 1 1 −1 1 1 1 1 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
72.30.1e1 C 1 1 1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
72.30.1e2 C 1 1 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1
72.30.1f1 C 1 1 −1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 ζ3 ζ3−1 1 ζ3 ζ3−1 −1 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3
72.30.1f2 C 1 1 −1 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 −1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1
72.30.1g1 C 1 1 −1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 −1 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3
72.30.1g2 C 1 1 −1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 −1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1
72.30.1h1 C 1 1 1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3
72.30.1h2 C 1 1 1 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1
72.30.2a R 2 2 2 0 2 2 −1 −1 −1 0 2 2 −1 −1 −1 −1 −1 2 2 −1 −1 −1 −1 0 0 0 0
72.30.2b R 2 −2 0 0 2 2 2 2 2 0 −2 −2 −2 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0
72.30.2c R 2 2 −2 0 2 2 −1 −1 −1 0 2 2 −1 −1 −1 1 1 −2 −2 1 1 1 1 0 0 0 0
72.30.2d1 C 2 2 2 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 −1 −1 2ζ3 2ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 0 0 0 0
72.30.2d2 C 2 2 2 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 −1 −1 2ζ3−1 2ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 0 0 0 0
72.30.2e1 C 2 −2 0 0 2 2 −1 −1 −1 0 −2 −2 1 1 1 −1−2ζ3 1+2ζ3 0 0 −1−2ζ3 1+2ζ3 1+2ζ3 −1−2ζ3 0 0 0 0
72.30.2e2 C 2 −2 0 0 2 2 −1 −1 −1 0 −2 −2 1 1 1 1+2ζ3 −1−2ζ3 0 0 1+2ζ3 −1−2ζ3 −1−2ζ3 1+2ζ3 0 0 0 0
72.30.2f1 C 2 −2 0 0 2ζ3−1 2ζ3 2 2ζ3−1 2ζ3 0 −2ζ3 −2ζ3−1 −2 −2ζ3 −2ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0
72.30.2f2 C 2 −2 0 0 2ζ3 2ζ3−1 2 2ζ3 2ζ3−1 0 −2ζ3−1 −2ζ3 −2 −2ζ3−1 −2ζ3 0 0 0 0 0 0 0 0 0 0 0 0
72.30.2g1 C 2 2 −2 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 1 1 −2ζ3 −2ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 0 0 0 0
72.30.2g2 C 2 2 −2 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 1 1 −2ζ3−1 −2ζ3 ζ3−1 ζ3 ζ3−1 ζ3 0 0 0 0
72.30.2h1 C 2 −2 0 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 0 −2ζ3 −2ζ3−1 1 ζ3 ζ3−1 1+2ζ3 −1−2ζ3 0 0 −2−ζ3 −1+ζ3 2+ζ3 1−ζ3 0 0 0 0
72.30.2h2 C 2 −2 0 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 0 −2ζ3−1 −2ζ3 1 ζ3−1 ζ3 −1−2ζ3 1+2ζ3 0 0 −1+ζ3 −2−ζ3 1−ζ3 2+ζ3 0 0 0 0
72.30.2i1 C 2 −2 0 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 0 −2ζ3 −2ζ3−1 1 ζ3 ζ3−1 −1−2ζ3 1+2ζ3 0 0 2+ζ3 1−ζ3 −2−ζ3 −1+ζ3 0 0 0 0
72.30.2i2 C 2 −2 0 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 0 −2ζ3−1 −2ζ3 1 ζ3−1 ζ3 1+2ζ3 −1−2ζ3 0 0 1−ζ3 2+ζ3 −1+ζ3 −2−ζ3 0 0 0 0

magma: CharacterTable(G);