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Properties

Label 36T20
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times S_4$

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Show commands: Magma

magma: G := TransitiveGroup(36, 20);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $20$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times S_4$
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:  (1,12,28,36,15,23,2,11,27,35,16,24)(3,10,25,34,14,22,4,9,26,33,13,21)(5,17,31,7,20,30)(6,18,32,8,19,29), (1,4)(2,3)(5,34)(6,33)(7,36)(8,35)(9,20)(10,19)(11,17)(12,18)(13,15)(14,16)(21,31)(22,32)(23,29)(24,30)(25,27)(26,28)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$
$24$:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$, $S_3$

Degree 4: None

Degree 6: $C_6$, $S_3$, $S_3\times C_3$, $S_4$, $S_4$

Degree 9: $S_3\times C_3$

Degree 12: $S_4$

Degree 18: $S_3 \times C_3$, 18T30, 18T33

Low degree siblings

12T45, 18T30, 18T33, 24T80, 24T84, 36T52

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

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:  72.42
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A1 3A-1 3B 3C1 3C-1 4A 6A1 6A-1 6B1 6B-1 12A1 12A-1
Size 1 3 6 1 1 8 8 8 6 3 3 6 6 6 6
2 P 1A 1A 1A 3A-1 3A1 3C-1 3C1 3B 2A 3A1 3A-1 3A1 3A-1 6A1 6A-1
3 P 1A 2A 2B 1A 1A 1A 1A 1A 4A 2A 2A 2B 2B 4A 4A
Type
72.42.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.42.1b R 1 1 −1 1 1 1 1 1 −1 1 1 −1 −1 −1 −1
72.42.1c1 C 1 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
72.42.1c2 C 1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
72.42.1d1 C 1 1 −1 ζ3−1 ζ3 1 ζ3 ζ3−1 −1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3
72.42.1d2 C 1 1 −1 ζ3 ζ3−1 1 ζ3−1 ζ3 −1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1
72.42.2a R 2 2 0 2 2 −1 −1 −1 0 2 2 0 0 0 0
72.42.2b1 C 2 2 0 2ζ3−1 2ζ3 −1 −ζ3 −ζ3−1 0 2ζ3 2ζ3−1 0 0 0 0
72.42.2b2 C 2 2 0 2ζ3 2ζ3−1 −1 −ζ3−1 −ζ3 0 2ζ3−1 2ζ3 0 0 0 0
72.42.3a R 3 −1 1 3 3 0 0 0 −1 −1 −1 1 1 −1 −1
72.42.3b R 3 −1 −1 3 3 0 0 0 1 −1 −1 −1 −1 1 1
72.42.3c1 C 3 −1 1 3ζ3−1 3ζ3 0 0 0 −1 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3
72.42.3c2 C 3 −1 1 3ζ3 3ζ3−1 0 0 0 −1 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1
72.42.3d1 C 3 −1 −1 3ζ3−1 3ζ3 0 0 0 1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3
72.42.3d2 C 3 −1 −1 3ζ3 3ζ3−1 0 0 0 1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1

magma: CharacterTable(G);