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Properties

Label 36T22
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_6^2:C_2$

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Show commands: Magma

magma: G := TransitiveGroup(36, 22);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $22$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_6^2:C_2$
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:  (1,6,2,5)(3,8,4,7)(9,24,10,23)(11,22,12,21)(13,30,14,29)(15,32,16,31)(17,26,18,25)(19,27,20,28)(33,36,34,35), (1,28,16,2,27,15)(3,26,13)(4,25,14)(5,32,19)(6,31,20)(7,30,17,8,29,18)(9,34,22,10,33,21)(11,35,23)(12,36,24), (1,10,29)(2,9,30)(3,12,31,4,11,32)(5,13,24,6,14,23)(7,16,21)(8,15,22)(17,27,34)(18,28,33)(19,26,36,20,25,35)
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 4
$8$:  $D_{4}$
$12$:  $D_{6}$ x 4
$18$:  $C_3^2:C_2$
$24$:  $(C_6\times C_2):C_2$ x 4
$36$:  18T12

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 4

Degree 4: $D_{4}$

Degree 6: $S_3$ x 4

Degree 9: $C_3^2:C_2$

Degree 12: $(C_6\times C_2):C_2$ x 4

Degree 18: $C_3^2 : C_2$

Low degree siblings

36T42

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 $2^{9},1^{18}$ $2$ $2$ $9$ $( 3, 4)( 5, 6)(11,12)(13,14)(19,20)(23,24)(25,26)(31,32)(35,36)$
2C $2^{18}$ $18$ $2$ $18$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,24)(10,23)(11,21)(12,22)(13,29)(14,30)(15,32)(16,31)(17,26)(18,25)(19,28)(20,27)(33,36)(34,35)$
3A $3^{12}$ $2$ $3$ $24$ $( 1,27,16)( 2,28,15)( 3,26,13)( 4,25,14)( 5,32,19)( 6,31,20)( 7,29,17)( 8,30,18)( 9,33,22)(10,34,21)(11,35,23)(12,36,24)$
3B $3^{12}$ $2$ $3$ $24$ $( 1,10,29)( 2, 9,30)( 3,11,31)( 4,12,32)( 5,14,24)( 6,13,23)( 7,16,21)( 8,15,22)(17,27,34)(18,28,33)(19,25,36)(20,26,35)$
3C $3^{12}$ $2$ $3$ $24$ $( 1,17,21)( 2,18,22)( 3,20,23)( 4,19,24)( 5,12,25)( 6,11,26)( 7,10,27)( 8, 9,28)(13,31,35)(14,32,36)(15,30,33)(16,29,34)$
3D $3^{12}$ $2$ $3$ $24$ $( 1,34, 7)( 2,33, 8)( 3,35, 6)( 4,36, 5)( 9,18,15)(10,17,16)(11,20,13)(12,19,14)(21,29,27)(22,30,28)(23,31,26)(24,32,25)$
4A $4^{9}$ $18$ $4$ $27$ $( 1,31, 2,32)( 3,30, 4,29)( 5,27, 6,28)( 7,26, 8,25)( 9,12,10,11)(13,18,14,17)(15,19,16,20)(21,35,22,36)(23,33,24,34)$
6A $6^{3},3^{6}$ $2$ $6$ $27$ $( 1, 7,34)( 2, 8,33)( 3, 5,35, 4, 6,36)( 9,15,18)(10,16,17)(11,14,20,12,13,19)(21,27,29)(22,28,30)(23,25,31,24,26,32)$
6B $6^{3},3^{6}$ $2$ $6$ $27$ $( 1, 8,34, 2, 7,33)( 3, 6,35)( 4, 5,36)( 9,16,18,10,15,17)(11,13,20)(12,14,19)(21,28,29,22,27,30)(23,26,31)(24,25,32)$
6C $6^{3},3^{6}$ $2$ $6$ $27$ $( 1,10,29)( 2, 9,30)( 3,12,31, 4,11,32)( 5,13,24, 6,14,23)( 7,16,21)( 8,15,22)(17,27,34)(18,28,33)(19,26,36,20,25,35)$
6D $6^{6}$ $2$ $6$ $30$ $( 1,18,21, 2,17,22)( 3,19,23, 4,20,24)( 5,11,25, 6,12,26)( 7, 9,27, 8,10,28)(13,32,35,14,31,36)(15,29,33,16,30,34)$
6E1 $6^{6}$ $2$ $6$ $30$ $( 1,28,16, 2,27,15)( 3,25,13, 4,26,14)( 5,31,19, 6,32,20)( 7,30,17, 8,29,18)( 9,34,22,10,33,21)(11,36,23,12,35,24)$
6E-1 $6^{3},3^{6}$ $2$ $6$ $27$ $( 1,15,27, 2,16,28)( 3,13,26)( 4,14,25)( 5,19,32)( 6,20,31)( 7,18,29, 8,17,30)( 9,21,33,10,22,34)(11,23,35)(12,24,36)$
6F1 $6^{3},3^{6}$ $2$ $6$ $27$ $( 1,29,10)( 2,30, 9)( 3,32,11, 4,31,12)( 5,23,14, 6,24,13)( 7,21,16)( 8,22,15)(17,34,27)(18,33,28)(19,35,25,20,36,26)$
6F-1 $6^{6}$ $2$ $6$ $30$ $( 1,33, 7, 2,34, 8)( 3,36, 6, 4,35, 5)( 9,17,15,10,18,16)(11,19,13,12,20,14)(21,30,27,22,29,28)(23,32,26,24,31,25)$
6G1 $6^{3},3^{6}$ $2$ $6$ $27$ $( 1,22,17, 2,21,18)( 3,23,20)( 4,24,19)( 5,25,12)( 6,26,11)( 7,28,10, 8,27, 9)(13,35,31)(14,36,32)(15,34,30,16,33,29)$
6G-1 $6^{6}$ $2$ $6$ $30$ $( 1,30,10, 2,29, 9)( 3,32,11, 4,31,12)( 5,23,14, 6,24,13)( 7,22,16, 8,21,15)(17,33,27,18,34,28)(19,35,25,20,36,26)$
6H1 $6^{3},3^{6}$ $2$ $6$ $27$ $( 1,16,27)( 2,15,28)( 3,14,26, 4,13,25)( 5,20,32, 6,19,31)( 7,17,29)( 8,18,30)( 9,22,33)(10,21,34)(11,24,35,12,23,36)$
6H-1 $6^{3},3^{6}$ $2$ $6$ $27$ $( 1,21,17)( 2,22,18)( 3,24,20, 4,23,19)( 5,26,12, 6,25,11)( 7,27,10)( 8,28, 9)(13,36,31,14,35,32)(15,33,30)(16,34,29)$

Malle's constant $a(G)$:     $1/9$

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:  72.35
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 3D 4A 6A 6B 6C 6D 6E1 6E-1 6F1 6F-1 6G1 6G-1 6H1 6H-1
Size 1 1 2 18 2 2 2 2 18 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 3A 3B 3C 3D 2A 3D 3D 3B 3C 3A 3A 3B 3D 3C 3B 3A 3C
3 P 1A 2A 2B 2C 1A 1A 1A 1A 4A 2B 2B 2B 2A 2A 2B 2B 2A 2B 2A 2B 2B
Type
72.35.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.35.1b R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
72.35.1c R 1 1 −1 1 1 1 1 1 −1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
72.35.1d R 1 1 1 −1 1 1 1 1 −1 1 1 1 1 1 1 1 1 1 1 1 1
72.35.2a R 2 2 2 0 −1 −1 −1 2 0 −1 −1 −1 2 −1 −1 −1 −1 −1 −1 2 2
72.35.2b R 2 2 2 0 −1 −1 2 −1 0 −1 −1 2 −1 −1 −1 −1 −1 2 2 −1 −1
72.35.2c R 2 2 2 0 −1 2 −1 −1 0 −1 2 −1 −1 −1 −1 2 2 −1 −1 −1 −1
72.35.2d R 2 2 2 0 2 −1 −1 −1 0 2 −1 −1 −1 2 2 −1 −1 −1 −1 −1 −1
72.35.2e R 2 −2 0 0 2 2 2 2 0 −2 −2 −2 −2 0 0 0 0 0 0 0 0
72.35.2f R 2 2 −2 0 −1 −1 −1 2 0 −1 −1 −1 2 1 1 1 1 1 1 −2 −2
72.35.2g R 2 2 −2 0 −1 −1 2 −1 0 −1 −1 2 −1 1 1 1 1 −2 −2 1 1
72.35.2h R 2 2 −2 0 −1 2 −1 −1 0 −1 2 −1 −1 1 1 −2 −2 1 1 1 1
72.35.2i R 2 2 −2 0 2 −1 −1 −1 0 2 −1 −1 −1 −2 −2 1 1 1 1 1 1
72.35.2j1 C 2 −2 0 0 −1 −1 −1 2 0 1 1 1 −2 −1−2ζ3 1+2ζ3 −1−2ζ3 1+2ζ3 −1−2ζ3 1+2ζ3 0 0
72.35.2j2 C 2 −2 0 0 −1 −1 −1 2 0 1 1 1 −2 1+2ζ3 −1−2ζ3 1+2ζ3 −1−2ζ3 1+2ζ3 −1−2ζ3 0 0
72.35.2k1 C 2 −2 0 0 −1 −1 2 −1 0 1 1 −2 1 −1−2ζ3 1+2ζ3 1+2ζ3 −1−2ζ3 0 0 1+2ζ3 −1−2ζ3
72.35.2k2 C 2 −2 0 0 −1 −1 2 −1 0 1 1 −2 1 1+2ζ3 −1−2ζ3 −1−2ζ3 1+2ζ3 0 0 −1−2ζ3 1+2ζ3
72.35.2l1 C 2 −2 0 0 −1 2 −1 −1 0 1 −2 1 1 −1−2ζ3 1+2ζ3 0 0 1+2ζ3 −1−2ζ3 −1−2ζ3 1+2ζ3
72.35.2l2 C 2 −2 0 0 −1 2 −1 −1 0 1 −2 1 1 1+2ζ3 −1−2ζ3 0 0 −1−2ζ3 1+2ζ3 1+2ζ3 −1−2ζ3
72.35.2m1 C 2 −2 0 0 2 −1 −1 −1 0 −2 1 1 1 0 0 −1−2ζ3 1+2ζ3 1+2ζ3 −1−2ζ3 1+2ζ3 −1−2ζ3
72.35.2m2 C 2 −2 0 0 2 −1 −1 −1 0 −2 1 1 1 0 0 1+2ζ3 −1−2ζ3 −1−2ζ3 1+2ζ3 −1−2ζ3 1+2ζ3

magma: CharacterTable(G);