↑

Properties

Label 36T5
Degree $36$
Order $36$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3:C_{12}$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(36, 5);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $5$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3:C_{12}$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $36$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,19,27,6,16,31,2,20,28,5,15,32)(3,17,26,7,13,29,4,18,25,8,14,30)(9,24,33,12,22,35,10,23,34,11,21,36), (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)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $S_3$, $C_6$
$12$:  $C_{12}$, $C_3 : C_4$
$18$:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$, $S_3$

Degree 4: $C_4$

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

Degree 9: $S_3\times C_3$

Degree 12: $C_{12}$, $C_3 : C_4$, $C_3\times (C_3 : C_4)$

Degree 18: $S_3 \times C_3$

Low degree siblings

12T19

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 $2^{18}$ $1$ $2$ $18$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
3A1 $3^{12}$ $1$ $3$ $24$ $( 1,16,28)( 2,15,27)( 3,13,25)( 4,14,26)( 5,19,31)( 6,20,32)( 7,18,30)( 8,17,29)( 9,22,34)(10,21,33)(11,24,35)(12,23,36)$
3A-1 $3^{12}$ $1$ $3$ $24$ $( 1,28,16)( 2,27,15)( 3,25,13)( 4,26,14)( 5,31,19)( 6,32,20)( 7,30,18)( 8,29,17)( 9,34,22)(10,33,21)(11,35,24)(12,36,23)$
3B $3^{12}$ $2$ $3$ $24$ $( 1,29, 9)( 2,30,10)( 3,32,12)( 4,31,11)( 5,24,14)( 6,23,13)( 7,21,15)( 8,22,16)(17,34,28)(18,33,27)(19,35,26)(20,36,25)$
3C1 $3^{12}$ $2$ $3$ $24$ $( 1, 8,34)( 2, 7,33)( 3, 6,36)( 4, 5,35)( 9,16,17)(10,15,18)(11,14,19)(12,13,20)(21,27,30)(22,28,29)(23,25,32)(24,26,31)$
3C-1 $3^{12}$ $2$ $3$ $24$ $( 1,17,22)( 2,18,21)( 3,20,23)( 4,19,24)( 5,11,26)( 6,12,25)( 7,10,27)( 8, 9,28)(13,32,36)(14,31,35)(15,30,33)(16,29,34)$
4A1 $4^{9}$ $3$ $4$ $27$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,12,10,11)(13,18,14,17)(15,19,16,20)(21,24,22,23)(25,30,26,29)(27,31,28,32)(33,35,34,36)$
4A-1 $4^{9}$ $3$ $4$ $27$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,11,10,12)(13,17,14,18)(15,20,16,19)(21,23,22,24)(25,29,26,30)(27,32,28,31)(33,36,34,35)$
6A1 $6^{6}$ $1$ $6$ $30$ $( 1,27,16, 2,28,15)( 3,26,13, 4,25,14)( 5,32,19, 6,31,20)( 7,29,18, 8,30,17)( 9,33,22,10,34,21)(11,36,24,12,35,23)$
6A-1 $6^{6}$ $1$ $6$ $30$ $( 1,15,28, 2,16,27)( 3,14,25, 4,13,26)( 5,20,31, 6,19,32)( 7,17,30, 8,18,29)( 9,21,34,10,22,33)(11,23,35,12,24,36)$
6B $6^{6}$ $2$ $6$ $30$ $( 1, 7,34, 2, 8,33)( 3, 5,36, 4, 6,35)( 9,15,17,10,16,18)(11,13,19,12,14,20)(21,28,30,22,27,29)(23,26,32,24,25,31)$
6C1 $6^{6}$ $2$ $6$ $30$ $( 1,30, 9, 2,29,10)( 3,31,12, 4,32,11)( 5,23,14, 6,24,13)( 7,22,15, 8,21,16)(17,33,28,18,34,27)(19,36,26,20,35,25)$
6C-1 $6^{6}$ $2$ $6$ $30$ $( 1,18,22, 2,17,21)( 3,19,23, 4,20,24)( 5,12,26, 6,11,25)( 7, 9,27, 8,10,28)(13,31,36,14,32,35)(15,29,33,16,30,34)$
12A1 $12^{3}$ $3$ $12$ $33$ $( 1,19,27, 6,16,31, 2,20,28, 5,15,32)( 3,17,26, 7,13,29, 4,18,25, 8,14,30)( 9,24,33,12,22,35,10,23,34,11,21,36)$
12A-1 $12^{3}$ $3$ $12$ $33$ $( 1,20,27, 5,16,32, 2,19,28, 6,15,31)( 3,18,26, 8,13,30, 4,17,25, 7,14,29)( 9,23,33,11,22,36,10,24,34,12,21,35)$
12A5 $12^{3}$ $3$ $12$ $33$ $( 1,32,15, 5,28,20, 2,31,16, 6,27,19)( 3,30,14, 8,25,18, 4,29,13, 7,26,17)( 9,36,21,11,34,23,10,35,22,12,33,24)$
12A-5 $12^{3}$ $3$ $12$ $33$ $( 1,31,15, 6,28,19, 2,32,16, 5,27,20)( 3,29,14, 7,25,17, 4,30,13, 8,26,18)( 9,35,21,12,34,24,10,36,22,11,33,23)$

Malle's constant $a(G)$:     $1/18$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  36.6
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B 3C1 3C-1 4A1 4A-1 6A1 6A-1 6B 6C1 6C-1 12A1 12A-1 12A5 12A-5
Size 1 1 1 1 2 2 2 3 3 1 1 2 2 2 3 3 3 3
2 P 1A 1A 3A-1 3A1 3C1 3B 3C-1 2A 2A 3A1 3A-1 3B 3C1 3C-1 6A1 6A1 6A-1 6A-1
3 P 1A 2A 1A 1A 1A 1A 1A 4A-1 4A1 2A 2A 2A 2A 2A 4A1 4A-1 4A-1 4A1
Type
36.6.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
36.6.1b R 1 1 1 1 1 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1
36.6.1c1 C 1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
36.6.1c2 C 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
36.6.1d1 C 1 −1 1 1 1 1 1 −i i −1 −1 −1 −1 −1 i −i i −i
36.6.1d2 C 1 −1 1 1 1 1 1 i −i −1 −1 −1 −1 −1 −i i −i i
36.6.1e1 C 1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 −1 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
36.6.1e2 C 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 −1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
36.6.1f1 C 1 −1 −ζ122 ζ124 1 −ζ122 ζ124 −ζ123 ζ123 −ζ124 ζ122 −1 −ζ124 ζ122 −ζ125 ζ12 −ζ12 ζ125
36.6.1f2 C 1 −1 ζ124 −ζ122 1 ζ124 −ζ122 ζ123 −ζ123 ζ122 −ζ124 −1 ζ122 −ζ124 ζ12 −ζ125 ζ125 −ζ12
36.6.1f3 C 1 −1 −ζ122 ζ124 1 −ζ122 ζ124 ζ123 −ζ123 −ζ124 ζ122 −1 −ζ124 ζ122 ζ125 −ζ12 ζ12 −ζ125
36.6.1f4 C 1 −1 ζ124 −ζ122 1 ζ124 −ζ122 −ζ123 ζ123 ζ122 −ζ124 −1 ζ122 −ζ124 −ζ12 ζ125 −ζ125 ζ12
36.6.2a R 2 2 2 2 −1 −1 −1 0 0 2 2 −1 −1 −1 0 0 0 0
36.6.2b S 2 −2 2 2 −1 −1 −1 0 0 −2 −2 1 1 1 0 0 0 0
36.6.2c1 C 2 2 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 0 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 0 0 0 0
36.6.2c2 C 2 2 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 0 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 0 0 0 0
36.6.2d1 C 2 −2 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 0 0 −2ζ3 −2ζ3−1 1 ζ3 ζ3−1 0 0 0 0
36.6.2d2 C 2 −2 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 0 0 −2ζ3−1 −2ζ3 1 ζ3−1 ζ3 0 0 0 0

magma: CharacterTable(G);