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Properties

Label 36T41
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{12}:S_3$

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Show commands: Magma

magma: G := TransitiveGroup(36, 41);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $41$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{12}:S_3$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,7)(2,8)(3,5)(4,6)(9,23)(10,24)(11,21)(12,22)(13,31)(14,32)(15,29)(16,30)(17,27)(18,28)(19,26)(20,25), (1,19,21)(2,20,22)(3,18,23)(4,17,24)(5,9,28)(6,10,27)(7,11,26)(8,12,25)(13,31,35)(14,32,36)(15,29,34)(16,30,33), (1,3,2,4)(5,35,6,36)(7,33,8,34)(9,31,10,32)(11,30,12,29)(13,27,14,28)(15,26,16,25)(17,21,18,22)(19,23,20,24)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$ x 4
$8$:  $C_4\times C_2$
$12$:  $D_{6}$ x 4
$18$:  $C_3^2:C_2$
$24$:  $S_3 \times C_4$ x 4
$36$:  18T12

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 4

Degree 4: $C_4$

Degree 6: $D_{6}$ x 4

Degree 9: $C_3^2:C_2$

Degree 12: $S_3 \times C_4$ x 4

Degree 18: 18T12

Low degree siblings

36T41

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

Malle's constant $a(G)$:     $1/16$

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:  72.32
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 3D 4A1 4A-1 4B1 4B-1 6A 6B 6C 6D 12A1 12A-1 12B1 12B-1 12C1 12C-1 12D1 12D-1
Size 1 1 9 9 2 2 2 2 1 1 9 9 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 3A 3D 3B 3C 2A 2A 2A 2A 3A 3B 3C 3D 6D 6A 6B 6B 6A 6C 6C 6D
3 P 1A 2A 2B 2C 1A 1A 1A 1A 4A-1 4A1 4B-1 4B1 2A 2A 2A 2A 4A1 4A1 4A1 4A-1 4A-1 4A1 4A-1 4A-1
Type
72.32.1a 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
72.32.1b 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
72.32.1c 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
72.32.1d 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
72.32.1e1 C 1 −1 −1 1 1 1 1 1 −i i i −i −1 −1 −1 −1 i −i −i i −i i −i i
72.32.1e2 C 1 −1 −1 1 1 1 1 1 i −i −i i −1 −1 −1 −1 −i i i −i i −i i −i
72.32.1f1 C 1 −1 1 −1 1 1 1 1 −i i −i i −1 −1 −1 −1 i −i −i i −i i −i i
72.32.1f2 C 1 −1 1 −1 1 1 1 1 i −i i −i −1 −1 −1 −1 −i i i −i i −i i −i
72.32.2a R 2 2 0 0 −1 −1 −1 2 2 2 0 0 −1 −1 −1 2 −1 −1 −1 −1 −1 −1 2 2
72.32.2b R 2 2 0 0 −1 −1 2 −1 2 2 0 0 −1 −1 2 −1 −1 −1 −1 −1 2 2 −1 −1
72.32.2c R 2 2 0 0 −1 2 −1 −1 2 2 0 0 −1 2 −1 −1 −1 −1 2 2 −1 −1 −1 −1
72.32.2d R 2 2 0 0 2 −1 −1 −1 2 2 0 0 2 −1 −1 −1 2 2 −1 −1 −1 −1 −1 −1
72.32.2e R 2 2 0 0 −1 −1 −1 2 −2 −2 0 0 −1 −1 −1 2 1 1 1 1 1 1 −2 −2
72.32.2f R 2 2 0 0 −1 −1 2 −1 −2 −2 0 0 −1 −1 2 −1 1 1 1 1 −2 −2 1 1
72.32.2g R 2 2 0 0 −1 2 −1 −1 −2 −2 0 0 −1 2 −1 −1 1 1 −2 −2 1 1 1 1
72.32.2h R 2 2 0 0 2 −1 −1 −1 −2 −2 0 0 2 −1 −1 −1 −2 −2 1 1 1 1 1 1
72.32.2i1 C 2 −2 0 0 −1 −1 −1 2 −2i 2i 0 0 1 1 1 −2 −i i i −i i −i −2i 2i
72.32.2i2 C 2 −2 0 0 −1 −1 −1 2 2i −2i 0 0 1 1 1 −2 i −i −i i −i i 2i −2i
72.32.2j1 C 2 −2 0 0 −1 −1 2 −1 −2i 2i 0 0 1 1 −2 1 −i i i −i −2i 2i i −i
72.32.2j2 C 2 −2 0 0 −1 −1 2 −1 2i −2i 0 0 1 1 −2 1 i −i −i i 2i −2i −i i
72.32.2k1 C 2 −2 0 0 −1 2 −1 −1 −2i 2i 0 0 1 −2 1 1 −i i −2i 2i i −i i −i
72.32.2k2 C 2 −2 0 0 −1 2 −1 −1 2i −2i 0 0 1 −2 1 1 i −i 2i −2i −i i −i i
72.32.2l1 C 2 −2 0 0 2 −1 −1 −1 −2i 2i 0 0 −2 1 1 1 2i −2i i −i i −i i −i
72.32.2l2 C 2 −2 0 0 2 −1 −1 −1 2i −2i 0 0 −2 1 1 1 −2i 2i −i i −i i −i i

magma: CharacterTable(G);