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Properties

Label 36T42
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, 42);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $42$
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);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,33,2,34)(3,35,4,36)(5,8,6,7)(9,25,10,26)(11,27,12,28)(13,23,14,24)(15,21,16,22)(17,32,18,31)(19,29,20,30), (1,24,20,4,22,18)(2,23,19,3,21,17)(5,26,10,8,28,11)(6,25,9,7,27,12)(13,34,31,16,36,29)(14,33,32,15,35,30), (1,12)(2,11)(3,9)(4,10)(5,18)(6,17)(7,20)(8,19)(13,36)(14,35)(15,34)(16,33)(21,26)(22,25)(23,27)(24,28)(29,30)
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: $D_{6}$ x 4

Degree 9: $C_3^2:C_2$

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

Degree 18: 18T12

Low degree siblings

36T22

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

Malle's constant $a(G)$:     $1/17$

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);