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Properties

Label 36T31
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_6\times A_4$

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Show commands: Magma

magma: G := TransitiveGroup(36, 31);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $31$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_6\times A_4$
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:  (1,20,30,2,19,29)(3,18,32,4,17,31)(5,22,33,6,21,34)(7,24,35,8,23,36)(9,15,27,10,16,28)(11,13,25,12,14,26), (1,15,36)(2,16,35)(3,14,33)(4,13,34)(5,20,26)(6,19,25)(7,17,28)(8,18,27)(9,22,32)(10,21,31)(11,24,29)(12,23,30)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$ x 4
$6$:  $C_6$ x 4
$9$:  $C_3^2$
$12$:  $A_4$
$18$:  $C_6 \times C_3$
$24$:  $A_4\times C_2$
$36$:  $C_3\times A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$ x 4

Degree 4: None

Degree 6: $A_4$, $A_4\times C_2$

Degree 9: $C_3^2$

Degree 12: $A_4\times C_2$

Degree 18: $A_4 \times C_3$, 18T25

Low degree siblings

18T25, 24T71 x 3, 36T18

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 $2^{18}$ $3$ $2$ $18$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32)(33,36)(34,35)$
2C $2^{12},1^{12}$ $3$ $2$ $12$ $(13,16)(14,15)(17,19)(18,20)(21,24)(22,23)(25,27)(26,28)(29,32)(30,31)(33,35)(34,36)$
3A1 $3^{12}$ $1$ $3$ $24$ $( 1, 9, 5)( 2,10, 6)( 3,12, 8)( 4,11, 7)(13,24,17)(14,23,18)(15,22,20)(16,21,19)(25,35,31)(26,36,32)(27,33,30)(28,34,29)$
3A-1 $3^{12}$ $1$ $3$ $24$ $( 1, 5, 9)( 2, 6,10)( 3, 8,12)( 4, 7,11)(13,17,24)(14,18,23)(15,20,22)(16,19,21)(25,31,35)(26,32,36)(27,30,33)(28,29,34)$
3B1 $3^{12}$ $4$ $3$ $24$ $( 1,30,19)( 2,29,20)( 3,32,17)( 4,31,18)( 5,33,21)( 6,34,22)( 7,35,23)( 8,36,24)( 9,27,16)(10,28,15)(11,25,14)(12,26,13)$
3B-1 $3^{12}$ $4$ $3$ $24$ $( 1,19,30)( 2,20,29)( 3,17,32)( 4,18,31)( 5,21,33)( 6,22,34)( 7,23,35)( 8,24,36)( 9,16,27)(10,15,28)(11,14,25)(12,13,26)$
3C1 $3^{12}$ $4$ $3$ $24$ $( 1,33,16)( 2,34,15)( 3,36,13)( 4,35,14)( 5,27,19)( 6,28,20)( 7,25,18)( 8,26,17)( 9,30,21)(10,29,22)(11,31,23)(12,32,24)$
3C-1 $3^{12}$ $4$ $3$ $24$ $( 1,21,27)( 2,22,28)( 3,24,26)( 4,23,25)( 5,16,30)( 6,15,29)( 7,14,31)( 8,13,32)( 9,19,33)(10,20,34)(11,18,35)(12,17,36)$
3D1 $3^{12}$ $4$ $3$ $24$ $( 1,16,33)( 2,15,34)( 3,13,36)( 4,14,35)( 5,19,27)( 6,20,28)( 7,18,25)( 8,17,26)( 9,21,30)(10,22,29)(11,23,31)(12,24,32)$
3D-1 $3^{12}$ $4$ $3$ $24$ $( 1,27,21)( 2,28,22)( 3,26,24)( 4,25,23)( 5,30,16)( 6,29,15)( 7,31,14)( 8,32,13)( 9,33,19)(10,34,20)(11,35,18)(12,36,17)$
6A1 $6^{6}$ $1$ $6$ $30$ $( 1,10, 5, 2, 9, 6)( 3,11, 8, 4,12, 7)(13,23,17,14,24,18)(15,21,20,16,22,19)(25,36,31,26,35,32)(27,34,30,28,33,29)$
6A-1 $6^{6}$ $1$ $6$ $30$ $( 1, 6, 9, 2, 5,10)( 3, 7,12, 4, 8,11)(13,18,24,14,17,23)(15,19,22,16,20,21)(25,32,35,26,31,36)(27,29,33,28,30,34)$
6B1 $6^{6}$ $3$ $6$ $30$ $( 1, 6, 9, 2, 5,10)( 3, 7,12, 4, 8,11)(13,20,24,15,17,22)(14,19,23,16,18,21)(25,29,35,28,31,34)(26,30,36,27,32,33)$
6B-1 $6^{6}$ $3$ $6$ $30$ $( 1,10, 5, 2, 9, 6)( 3,11, 8, 4,12, 7)(13,22,17,15,24,20)(14,21,18,16,23,19)(25,34,31,28,35,29)(26,33,32,27,36,30)$
6C1 $6^{4},3^{4}$ $3$ $6$ $28$ $( 1, 5, 9)( 2, 6,10)( 3, 8,12)( 4, 7,11)(13,19,24,16,17,21)(14,20,23,15,18,22)(25,30,35,27,31,33)(26,29,36,28,32,34)$
6C-1 $6^{4},3^{4}$ $3$ $6$ $28$ $( 1, 9, 5)( 2,10, 6)( 3,12, 8)( 4,11, 7)(13,21,17,16,24,19)(14,22,18,15,23,20)(25,33,31,27,35,30)(26,34,32,28,36,29)$
6D1 $6^{6}$ $4$ $6$ $30$ $( 1,20,30, 2,19,29)( 3,18,32, 4,17,31)( 5,22,33, 6,21,34)( 7,24,35, 8,23,36)( 9,15,27,10,16,28)(11,13,25,12,14,26)$
6D-1 $6^{6}$ $4$ $6$ $30$ $( 1,28,21, 2,27,22)( 3,25,24, 4,26,23)( 5,29,16, 6,30,15)( 7,32,14, 8,31,13)( 9,34,19,10,33,20)(11,36,18,12,35,17)$
6E1 $6^{6}$ $4$ $6$ $30$ $( 1,22,27, 2,21,28)( 3,23,26, 4,24,25)( 5,15,30, 6,16,29)( 7,13,31, 8,14,32)( 9,20,33,10,19,34)(11,17,35,12,18,36)$
6E-1 $6^{6}$ $4$ $6$ $30$ $( 1,15,33, 2,16,34)( 3,14,36, 4,13,35)( 5,20,27, 6,19,28)( 7,17,25, 8,18,26)( 9,22,30,10,21,29)(11,24,31,12,23,32)$
6F1 $6^{6}$ $4$ $6$ $30$ $( 1,29,19, 2,30,20)( 3,31,17, 4,32,18)( 5,34,21, 6,33,22)( 7,36,23, 8,35,24)( 9,28,16,10,27,15)(11,26,14,12,25,13)$
6F-1 $6^{6}$ $4$ $6$ $30$ $( 1,34,16, 2,33,15)( 3,35,13, 4,36,14)( 5,28,19, 6,27,20)( 7,26,18, 8,25,17)( 9,29,21,10,30,22)(11,32,23,12,31,24)$

Malle's constant $a(G)$:     $1/12$

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:  72.47
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 6D1 6D-1 6E1 6E-1 6F1 6F-1
Size 1 1 3 3 1 1 4 4 4 4 4 4 1 1 3 3 3 3 4 4 4 4 4 4
2 P 1A 1A 1A 1A 3A-1 3A1 3B-1 3B1 3D-1 3C1 3D1 3C-1 3A-1 3A1 3A1 3A-1 3A1 3A-1 3B1 3C-1 3C1 3D1 3B-1 3D-1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2B 2B 2C 2C 2A 2A 2A 2A 2A 2A
Type
72.47.1a 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
72.47.1b 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
72.47.1c1 C 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1
72.47.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1
72.47.1d1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3 ζ3−1
72.47.1d2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3−1 ζ3
72.47.1e1 C 1 1 1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 ζ3
72.47.1e2 C 1 1 1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 ζ3−1
72.47.1f1 C 1 1 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1
72.47.1f2 C 1 1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3
72.47.1g1 C 1 −1 −1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −1 −1
72.47.1g2 C 1 −1 −1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −1 −1
72.47.1h1 C 1 −1 −1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −1 −1 −ζ3 −ζ3−1
72.47.1h2 C 1 −1 −1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −1 −1 −ζ3−1 −ζ3
72.47.1i1 C 1 −1 −1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 −1 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3
72.47.1i2 C 1 −1 −1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 −1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
72.47.1j1 C 1 −1 −1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 −1 −1 −1 −1 1 1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1
72.47.1j2 C 1 −1 −1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 −1 −1 −1 −1 1 1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3
72.47.3a R 3 3 −1 −1 3 3 0 0 0 0 0 0 3 3 −1 −1 −1 −1 0 0 0 0 0 0
72.47.3b R 3 −3 1 −1 3 3 0 0 0 0 0 0 −3 −3 1 1 −1 −1 0 0 0 0 0 0
72.47.3c1 C 3 3 −1 −1 3ζ3−1 3ζ3 0 0 0 0 0 0 3ζ3 3ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 0 0 0 0 0 0
72.47.3c2 C 3 3 −1 −1 3ζ3 3ζ3−1 0 0 0 0 0 0 3ζ3−1 3ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 0 0 0 0 0 0
72.47.3d1 C 3 −3 1 −1 3ζ3−1 3ζ3 0 0 0 0 0 0 −3ζ3 −3ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3 0 0 0 0 0 0
72.47.3d2 C 3 −3 1 −1 3ζ3 3ζ3−1 0 0 0 0 0 0 −3ζ3−1 −3ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1 0 0 0 0 0 0

magma: CharacterTable(G);