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Properties

Label 36T18
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, 18);
 

Group action invariants

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

Degree 3: $C_3$ x 4

Degree 4: None

Degree 6: $C_6$ x 4, $A_4$, $A_4\times C_2$

Degree 9: $C_3^2$

Degree 12: $A_4 \times C_2$

Degree 18: $C_6 \times C_3$, $A_4 \times C_3$, 18T25

Low degree siblings

18T25, 24T71 x 3, 36T31

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

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);