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Properties

Label 36T27
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times C_{12}$

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Show commands: Magma

magma: G := TransitiveGroup(36, 27);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$8$:  $C_4\times C_2$
$12$:  $D_{6}$, $C_{12}$ x 2, $C_6\times C_2$
$18$:  $S_3\times C_3$
$24$:  $S_3 \times C_4$, 24T2
$36$:  $C_6\times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$, $S_3$

Degree 4: $C_4$

Degree 6: $C_6$, $D_{6}$

Degree 9: $S_3\times C_3$

Degree 12: $C_{12}$, $S_3 \times C_4$

Degree 18: $S_3 \times C_6$

Low degree siblings

24T65, 36T27

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

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:  72.27
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);