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Properties

Label 36T35
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times C_3^2:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(36, 35);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $35$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times C_3^2:C_4$
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:  (1,13,28,2,14,27)(3,15,25,4,16,26)(5,20,32,6,19,31)(7,17,30,8,18,29)(9,21,36,10,22,35)(11,24,33,12,23,34), (1,25,33,9)(2,26,34,10)(3,27,36,12)(4,28,35,11)(5,7,6,8)(13,17,24,30)(14,18,23,29)(15,19,21,31)(16,20,22,32)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$36$:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: $C_3^2:C_4$ x 2

Degree 9: $C_3^2:C_4$

Degree 12: 12T41 x 2

Degree 18: $C_3^2 : C_4$

Low degree siblings

12T40 x 2, 12T41 x 2, 18T27 x 2, 24T76 x 2, 36T36

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

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:  72.45
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 4A1 4A-1 4B1 4B-1 6A 6B
Size 1 1 9 9 4 4 9 9 9 9 4 4
2 P 1A 1A 1A 1A 3A 3B 2B 2B 2B 2B 3A 3B
3 P 1A 2A 2B 2C 1A 1A 4B-1 4A-1 4A1 4B1 2A 2A
Type
72.45.1a R 1 1 1 1 1 1 1 1 1 1 1 1
72.45.1b R 1 −1 1 −1 1 1 −1 −1 1 1 −1 −1
72.45.1c R 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1
72.45.1d R 1 1 1 1 1 1 −1 −1 −1 −1 1 1
72.45.1e1 C 1 −1 −1 1 1 1 −i i −i i −1 −1
72.45.1e2 C 1 −1 −1 1 1 1 i −i i −i −1 −1
72.45.1f1 C 1 1 −1 −1 1 1 −i i i −i 1 1
72.45.1f2 C 1 1 −1 −1 1 1 i −i −i i 1 1
72.45.4a R 4 4 0 0 −2 1 0 0 0 0 −2 1
72.45.4b R 4 4 0 0 1 −2 0 0 0 0 1 −2
72.45.4c R 4 −4 0 0 −2 1 0 0 0 0 2 −1
72.45.4d R 4 −4 0 0 1 −2 0 0 0 0 −1 2

magma: CharacterTable(G);