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Properties

Label 36T26
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, 26);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $26$
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);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,31,11,3,29,10)(2,32,12,4,30,9)(5,22,14,8,24,16)(6,21,13,7,23,15)(17,33,25,20,35,27)(18,34,26,19,36,28), (1,24,15,34,25,10,2,23,16,33,26,9)(3,21,13,35,27,12,4,22,14,36,28,11)(5,30,19,8,31,18,6,29,20,7,32,17)
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: $C_6$, $D_{6}$

Degree 9: $S_3\times C_3$

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

Degree 18: $S_3 \times C_6$

Low degree siblings

12T42 x 2, 24T77, 36T19

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

Malle's constant $a(G)$:     $1/15$

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);