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Properties

Label 36T19
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_6\wr C_2$

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Show commands: Magma

magma: G := TransitiveGroup(36, 19);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$8$:  $D_{4}$
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$24$:  $(C_6\times C_2):C_2$, $D_4 \times C_3$
$36$:  $C_6\times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$, $S_3$

Degree 4: $D_{4}$

Degree 6: $C_6$, $S_3$, $S_3\times C_3$

Degree 9: $S_3\times C_3$

Degree 12: $D_4 \times C_3$, $(C_6\times C_2):C_2$, 12T42 x 2

Degree 18: $S_3 \times C_3$

Low degree siblings

12T42 x 2, 24T77, 36T26

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 $2^{9},1^{18}$ $2$ $2$ $9$ $( 1, 2)( 7, 8)( 9,10)(15,16)(17,18)(21,22)(27,28)(29,30)(33,34)$
2C $2^{18}$ $6$ $2$ $18$ $( 1,36)( 2,35)( 3,34)( 4,33)( 5, 7)( 6, 8)( 9,14)(10,13)(11,15)(12,16)(17,19)(18,20)(21,25)(22,26)(23,28)(24,27)(29,31)(30,32)$
3A1 $3^{12}$ $1$ $3$ $24$ $( 1,27,15)( 2,28,16)( 3,25,14)( 4,26,13)( 5,32,19)( 6,31,20)( 7,30,17)( 8,29,18)( 9,34,21)(10,33,22)(11,36,24)(12,35,23)$
3A-1 $3^{12}$ $1$ $3$ $24$ $( 1,15,27)( 2,16,28)( 3,14,25)( 4,13,26)( 5,19,32)( 6,20,31)( 7,17,30)( 8,18,29)( 9,21,34)(10,22,33)(11,24,36)(12,23,35)$
3B $3^{12}$ $2$ $3$ $24$ $( 1, 7,33)( 2, 8,34)( 3, 6,35)( 4, 5,36)( 9,16,18)(10,15,17)(11,13,19)(12,14,20)(21,28,29)(22,27,30)(23,25,31)(24,26,32)$
3C1 $3^{12}$ $2$ $3$ $24$ $( 1,17,22)( 2,18,21)( 3,20,23)( 4,19,24)( 5,11,26)( 6,12,25)( 7,10,27)( 8, 9,28)(13,32,36)(14,31,35)(15,30,33)(16,29,34)$
3C-1 $3^{12}$ $2$ $3$ $24$ $( 1,22,17)( 2,21,18)( 3,23,20)( 4,24,19)( 5,26,11)( 6,25,12)( 7,27,10)( 8,28, 9)(13,36,32)(14,35,31)(15,33,30)(16,34,29)$
4A $4^{9}$ $6$ $4$ $27$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,12,10,11)(13,18,14,17)(15,19,16,20)(21,23,22,24)(25,30,26,29)(27,32,28,31)(33,36,34,35)$
6A1 $6^{6}$ $1$ $6$ $30$ $( 1,16,27, 2,15,28)( 3,13,25, 4,14,26)( 5,20,32, 6,19,31)( 7,18,30, 8,17,29)( 9,22,34,10,21,33)(11,23,36,12,24,35)$
6A-1 $6^{6}$ $1$ $6$ $30$ $( 1,28,15, 2,27,16)( 3,26,14, 4,25,13)( 5,31,19, 6,32,20)( 7,29,17, 8,30,18)( 9,33,21,10,34,22)(11,35,24,12,36,23)$
6B $6^{6}$ $2$ $6$ $30$ $( 1,34, 7, 2,33, 8)( 3,36, 6, 4,35, 5)( 9,17,16,10,18,15)(11,20,13,12,19,14)(21,30,28,22,29,27)(23,32,25,24,31,26)$
6C1 $6^{3},3^{6}$ $2$ $6$ $27$ $( 1,27,15)( 2,28,16)( 3,26,14, 4,25,13)( 5,31,19, 6,32,20)( 7,30,17)( 8,29,18)( 9,34,21)(10,33,22)(11,35,24,12,36,23)$
6C-1 $6^{3},3^{6}$ $2$ $6$ $27$ $( 1,15,27)( 2,16,28)( 3,13,25, 4,14,26)( 5,20,32, 6,19,31)( 7,17,30)( 8,18,29)( 9,21,34)(10,22,33)(11,23,36,12,24,35)$
6D1 $6^{3},3^{6}$ $2$ $6$ $27$ $( 1,29,10, 2,30, 9)( 3,31,12)( 4,32,11)( 5,24,13)( 6,23,14)( 7,21,15, 8,22,16)(17,34,27,18,33,28)(19,36,26)(20,35,25)$
6D-1 $6^{3},3^{6}$ $2$ $6$ $27$ $( 1, 9,30, 2,10,29)( 3,12,31)( 4,11,32)( 5,13,24)( 6,14,23)( 7,16,22, 8,15,21)(17,28,33,18,27,34)(19,26,36)(20,25,35)$
6E1 $6^{6}$ $2$ $6$ $30$ $( 1,29,10, 2,30, 9)( 3,32,12, 4,31,11)( 5,23,13, 6,24,14)( 7,21,15, 8,22,16)(17,34,27,18,33,28)(19,35,26,20,36,25)$
6E-1 $6^{6}$ $2$ $6$ $30$ $( 1, 9,30, 2,10,29)( 3,11,31, 4,12,32)( 5,14,24, 6,13,23)( 7,16,22, 8,15,21)(17,28,33,18,27,34)(19,25,36,20,26,35)$
6F1 $6^{3},3^{6}$ $2$ $6$ $27$ $( 1,33, 7)( 2,34, 8)( 3,36, 6, 4,35, 5)( 9,18,16)(10,17,15)(11,20,13,12,19,14)(21,29,28)(22,30,27)(23,32,25,24,31,26)$
6F-1 $6^{3},3^{6}$ $2$ $6$ $27$ $( 1, 7,33)( 2, 8,34)( 3, 5,35, 4, 6,36)( 9,16,18)(10,15,17)(11,14,19,12,13,20)(21,28,29)(22,27,30)(23,26,31,24,25,32)$
6G1 $6^{3},3^{6}$ $2$ $6$ $27$ $( 1,18,22, 2,17,21)( 3,20,23)( 4,19,24)( 5,11,26)( 6,12,25)( 7, 9,27, 8,10,28)(13,32,36)(14,31,35)(15,29,33,16,30,34)$
6G-1 $6^{3},3^{6}$ $2$ $6$ $27$ $( 1,21,17, 2,22,18)( 3,23,20)( 4,24,19)( 5,26,11)( 6,25,12)( 7,28,10, 8,27, 9)(13,36,32)(14,35,31)(15,34,30,16,33,29)$
6H1 $6^{6}$ $6$ $6$ $30$ $( 1,13,27, 4,15,26)( 2,14,28, 3,16,25)( 5,10,32,33,19,22)( 6, 9,31,34,20,21)( 7,11,30,36,17,24)( 8,12,29,35,18,23)$
6H-1 $6^{6}$ $6$ $6$ $30$ $( 1,26,15, 4,27,13)( 2,25,16, 3,28,14)( 5,22,19,33,32,10)( 6,21,20,34,31, 9)( 7,24,17,36,30,11)( 8,23,18,35,29,12)$
12A1 $12^{3}$ $6$ $12$ $33$ $( 1,31,16, 5,27,20, 2,32,15, 6,28,19)( 3,29,13, 7,25,18, 4,30,14, 8,26,17)( 9,36,22,12,34,24,10,35,21,11,33,23)$
12A-1 $12^{3}$ $6$ $12$ $33$ $( 1,20,28, 5,15,31, 2,19,27, 6,16,32)( 3,18,26, 7,14,29, 4,17,25, 8,13,30)( 9,24,33,12,21,36,10,23,34,11,22,35)$

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:  72.30
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 3B 3C1 3C-1 4A 6A1 6A-1 6B 6C1 6C-1 6D1 6D-1 6E1 6E-1 6F1 6F-1 6G1 6G-1 6H1 6H-1 12A1 12A-1
Size 1 1 2 6 1 1 2 2 2 6 1 1 2 2 2 2 2 2 2 2 2 2 2 6 6 6 6
2 P 1A 1A 1A 1A 3A-1 3A1 3B 3C-1 3C1 2A 3A1 3A-1 3B 3A-1 3A1 3C1 3C-1 3C1 3C-1 3B 3B 3C-1 3C1 3A1 3A-1 6A1 6A-1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 4A 2A 2A 2A 2B 2B 2B 2B 2A 2A 2B 2B 2B 2B 2C 2C 4A 4A
Type
72.30.1a 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 1 1 1
72.30.1b 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 −1 1 1
72.30.1c 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 1 −1 −1
72.30.1d 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 −1 −1 −1
72.30.1e1 C 1 1 1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
72.30.1e2 C 1 1 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1
72.30.1f1 C 1 1 −1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 ζ3 ζ3−1 1 ζ3 ζ3−1 −1 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3
72.30.1f2 C 1 1 −1 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 −1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1
72.30.1g1 C 1 1 −1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 −1 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3
72.30.1g2 C 1 1 −1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 −1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1
72.30.1h1 C 1 1 1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3
72.30.1h2 C 1 1 1 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1
72.30.2a R 2 2 2 0 2 2 −1 −1 −1 0 2 2 −1 −1 −1 −1 −1 2 2 −1 −1 −1 −1 0 0 0 0
72.30.2b R 2 −2 0 0 2 2 2 2 2 0 −2 −2 −2 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0
72.30.2c R 2 2 −2 0 2 2 −1 −1 −1 0 2 2 −1 −1 −1 1 1 −2 −2 1 1 1 1 0 0 0 0
72.30.2d1 C 2 2 2 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 −1 −1 2ζ3 2ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 0 0 0 0
72.30.2d2 C 2 2 2 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 −1 −1 2ζ3−1 2ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 0 0 0 0
72.30.2e1 C 2 −2 0 0 2 2 −1 −1 −1 0 −2 −2 1 1 1 −1−2ζ3 1+2ζ3 0 0 −1−2ζ3 1+2ζ3 1+2ζ3 −1−2ζ3 0 0 0 0
72.30.2e2 C 2 −2 0 0 2 2 −1 −1 −1 0 −2 −2 1 1 1 1+2ζ3 −1−2ζ3 0 0 1+2ζ3 −1−2ζ3 −1−2ζ3 1+2ζ3 0 0 0 0
72.30.2f1 C 2 −2 0 0 2ζ3−1 2ζ3 2 2ζ3−1 2ζ3 0 −2ζ3 −2ζ3−1 −2 −2ζ3 −2ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0
72.30.2f2 C 2 −2 0 0 2ζ3 2ζ3−1 2 2ζ3 2ζ3−1 0 −2ζ3−1 −2ζ3 −2 −2ζ3−1 −2ζ3 0 0 0 0 0 0 0 0 0 0 0 0
72.30.2g1 C 2 2 −2 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 1 1 −2ζ3 −2ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 0 0 0 0
72.30.2g2 C 2 2 −2 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 1 1 −2ζ3−1 −2ζ3 ζ3−1 ζ3 ζ3−1 ζ3 0 0 0 0
72.30.2h1 C 2 −2 0 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 0 −2ζ3 −2ζ3−1 1 ζ3 ζ3−1 1+2ζ3 −1−2ζ3 0 0 −2−ζ3 −1+ζ3 2+ζ3 1−ζ3 0 0 0 0
72.30.2h2 C 2 −2 0 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 0 −2ζ3−1 −2ζ3 1 ζ3−1 ζ3 −1−2ζ3 1+2ζ3 0 0 −1+ζ3 −2−ζ3 1−ζ3 2+ζ3 0 0 0 0
72.30.2i1 C 2 −2 0 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 0 −2ζ3 −2ζ3−1 1 ζ3 ζ3−1 −1−2ζ3 1+2ζ3 0 0 2+ζ3 1−ζ3 −2−ζ3 −1+ζ3 0 0 0 0
72.30.2i2 C 2 −2 0 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 0 −2ζ3−1 −2ζ3 1 ζ3−1 ζ3 1+2ζ3 −1−2ζ3 0 0 1−ζ3 2+ζ3 −1+ζ3 −2−ζ3 0 0 0 0

magma: CharacterTable(G);