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Properties

Label 36T28
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times D_{12}$

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Show commands: Magma

magma: G := TransitiveGroup(36, 28);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $28$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times D_{12}$
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:  (1,36)(2,35)(3,34)(4,33)(7,8)(9,14)(10,13)(11,15)(12,16)(17,18)(21,26)(22,25)(23,28)(24,27)(29,30), (1,32,11,3,29,9,2,31,12,4,30,10)(5,22,13,7,24,16,6,21,14,8,23,15)(17,33,26,19,35,27,18,34,25,20,36,28)
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
$8$:  $D_{4}$
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$24$:  $D_{12}$, $D_4 \times C_3$
$36$:  $C_6\times S_3$

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: $D_{12}$, $D_4 \times C_3$

Degree 18: $S_3 \times C_6$

Low degree siblings

24T67, 36T28

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 $2^{18}$ $6$ $2$ $18$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,11)(10,12)(13,17)(14,18)(15,19)(16,20)(21,23)(22,24)(25,31)(26,32)(27,30)(28,29)(33,36)(34,35)$
2C $2^{15},1^{6}$ $6$ $2$ $15$ $( 1,36)( 2,35)( 3,34)( 4,33)( 7, 8)( 9,14)(10,13)(11,15)(12,16)(17,18)(21,26)(22,25)(23,28)(24,27)(29,30)$
3A1 $3^{12}$ $1$ $3$ $24$ $( 1,15,25)( 2,16,26)( 3,13,28)( 4,14,27)( 5,20,32)( 6,19,31)( 7,17,29)( 8,18,30)( 9,24,33)(10,23,34)(11,22,36)(12,21,35)$
3A-1 $3^{12}$ $1$ $3$ $24$ $( 1,25,15)( 2,26,16)( 3,28,13)( 4,27,14)( 5,32,20)( 6,31,19)( 7,29,17)( 8,30,18)( 9,33,24)(10,34,23)(11,36,22)(12,35,21)$
3B $3^{12}$ $2$ $3$ $24$ $( 1, 7,35)( 2, 8,36)( 3, 6,34)( 4, 5,33)( 9,14,20)(10,13,19)(11,16,18)(12,15,17)(21,25,29)(22,26,30)(23,28,31)(24,27,32)$
3C1 $3^{12}$ $2$ $3$ $24$ $( 1,17,21)( 2,18,22)( 3,19,23)( 4,20,24)( 5, 9,27)( 6,10,28)( 7,12,25)( 8,11,26)(13,31,34)(14,32,33)(15,29,35)(16,30,36)$
3C-1 $3^{12}$ $2$ $3$ $24$ $( 1,29,12)( 2,30,11)( 3,31,10)( 4,32, 9)( 5,24,14)( 6,23,13)( 7,21,15)( 8,22,16)(17,35,25)(18,36,26)(19,34,28)(20,33,27)$
4A $4^{9}$ $2$ $4$ $27$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,16,14,15)(17,19,18,20)(21,23,22,24)(25,28,26,27)(29,31,30,32)(33,35,34,36)$
6A1 $6^{6}$ $1$ $6$ $30$ $( 1,26,15, 2,25,16)( 3,27,13, 4,28,14)( 5,31,20, 6,32,19)( 7,30,17, 8,29,18)( 9,34,24,10,33,23)(11,35,22,12,36,21)$
6A-1 $6^{6}$ $1$ $6$ $30$ $( 1,16,25, 2,15,26)( 3,14,28, 4,13,27)( 5,19,32, 6,20,31)( 7,18,29, 8,17,30)( 9,23,33,10,24,34)(11,21,36,12,22,35)$
6B $6^{6}$ $2$ $6$ $30$ $( 1,22,17, 2,21,18)( 3,24,19, 4,23,20)( 5,28, 9, 6,27,10)( 7,26,12, 8,25,11)(13,33,31,14,34,32)(15,36,29,16,35,30)$
6C1 $6^{6}$ $2$ $6$ $30$ $( 1,11,29, 2,12,30)( 3, 9,31, 4,10,32)( 5,13,24, 6,14,23)( 7,16,21, 8,15,22)(17,26,35,18,25,36)(19,27,34,20,28,33)$
6C-1 $6^{6}$ $2$ $6$ $30$ $( 1,36, 7, 2,35, 8)( 3,33, 6, 4,34, 5)( 9,19,14,10,20,13)(11,17,16,12,18,15)(21,30,25,22,29,26)(23,32,28,24,31,27)$
6D1 $6^{6}$ $6$ $6$ $30$ $( 1,31,15, 6,25,19)( 2,32,16, 5,26,20)( 3,29,13, 7,28,17)( 4,30,14, 8,27,18)( 9,36,24,11,33,22)(10,35,23,12,34,21)$
6D-1 $6^{6}$ $6$ $6$ $30$ $( 1,19,25, 6,15,31)( 2,20,26, 5,16,32)( 3,17,28, 7,13,29)( 4,18,27, 8,14,30)( 9,22,33,11,24,36)(10,21,34,12,23,35)$
6E1 $6^{5},3^{2}$ $6$ $6$ $29$ $( 1,22,15,36,25,11)( 2,21,16,35,26,12)( 3,23,13,34,28,10)( 4,24,14,33,27, 9)( 5,32,20)( 6,31,19)( 7,30,17, 8,29,18)$
6E-1 $6^{5},3^{2}$ $6$ $6$ $29$ $( 1,11,25,36,15,22)( 2,12,26,35,16,21)( 3,10,28,34,13,23)( 4, 9,27,33,14,24)( 5,20,32)( 6,19,31)( 7,18,29, 8,17,30)$
12A1 $12^{3}$ $2$ $12$ $33$ $( 1,10,30, 4,12,31, 2, 9,29, 3,11,32)( 5,15,23, 8,14,21, 6,16,24, 7,13,22)(17,28,36,20,25,34,18,27,35,19,26,33)$
12A-1 $12^{3}$ $2$ $12$ $33$ $( 1,28,16, 4,25,13, 2,27,15, 3,26,14)( 5,29,19, 8,32,17, 6,30,20, 7,31,18)( 9,35,23,11,33,21,10,36,24,12,34,22)$
12B1 $12^{3}$ $2$ $12$ $33$ $( 1,23,18, 4,21,19, 2,24,17, 3,22,20)( 5,25,10, 8,27,12, 6,26, 9, 7,28,11)(13,36,32,15,34,30,14,35,31,16,33,29)$
12B5 $12^{3}$ $2$ $12$ $33$ $( 1,34, 8, 4,35, 6, 2,33, 7, 3,36, 5)( 9,17,13,11,20,15,10,18,14,12,19,16)(21,31,26,24,29,28,22,32,25,23,30,27)$
12C1 $12^{3}$ $2$ $12$ $33$ $( 1,13,26, 4,15,28, 2,14,25, 3,16,27)( 5,17,31, 8,20,29, 6,18,32, 7,19,30)( 9,21,34,11,24,35,10,22,33,12,23,36)$
12C-1 $12^{3}$ $2$ $12$ $33$ $( 1, 6,36, 4, 7,34, 2, 5,35, 3, 8,33)( 9,15,19,11,14,17,10,16,20,12,13,18)(21,28,30,24,25,31,22,27,29,23,26,32)$
12C5 $12^{3}$ $2$ $12$ $33$ $( 1,31,11, 4,29,10, 2,32,12, 3,30, 9)( 5,21,13, 8,24,15, 6,22,14, 7,23,16)(17,34,26,20,35,28,18,33,25,19,36,27)$
12C-5 $12^{3}$ $2$ $12$ $33$ $( 1,19,22, 4,17,23, 2,20,21, 3,18,24)( 5,12,28, 8, 9,25, 6,11,27, 7,10,26)(13,30,33,15,31,36,14,29,34,16,32,35)$

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:  72.28
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 12A1 12A-1 12B1 12B5 12C1 12C-1 12C5 12C-5
Size 1 1 6 6 1 1 2 2 2 2 1 1 2 2 2 6 6 6 6 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 3A-1 3A1 3B 3C-1 3C1 2A 3A1 3A-1 3C1 3C-1 3B 3A1 3A-1 3A1 3A-1 6C1 6A-1 6C-1 6B 6A1 6B 6C-1 6C1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 4A 2A 2A 2A 2A 2A 2B 2B 2C 2C 4A 4A 4A 4A 4A 4A 4A 4A
Type
72.28.1a 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
72.28.1b 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
72.28.1c 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
72.28.1d 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
72.28.1e1 C 1 1 1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
72.28.1e2 C 1 1 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
72.28.1f1 C 1 1 −1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
72.28.1f2 C 1 1 −1 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
72.28.1g1 C 1 1 −1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 −1 ζ3−1 ζ3 1 ζ3 ζ3−1 −ζ3 −ζ3−1 ζ3 ζ3−1 −1 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
72.28.1g2 C 1 1 −1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 −1 ζ3 ζ3−1 1 ζ3−1 ζ3 −ζ3−1 −ζ3 ζ3−1 ζ3 −1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
72.28.1h1 C 1 1 1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 −1 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1 −1 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
72.28.1h2 C 1 1 1 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 −1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3 −1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
72.28.2a R 2 2 0 0 2 2 −1 −1 −1 2 2 2 −1 −1 −1 0 0 0 0 −1 −1 2 2 −1 −1 −1 −1
72.28.2b 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
72.28.2c R 2 2 0 0 2 2 −1 −1 −1 −2 2 2 −1 −1 −1 0 0 0 0 1 1 −2 −2 1 1 1 1
72.28.2d1 C 2 2 0 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 2 2ζ3−1 2ζ3 −1 −ζ3 −ζ3−1 0 0 0 0 −1 −1 2ζ3 2ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
72.28.2d2 C 2 2 0 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 2 2ζ3 2ζ3−1 −1 −ζ3−1 −ζ3 0 0 0 0 −1 −1 2ζ3−1 2ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
72.28.2e1 R 2 −2 0 0 2 2 −1 −1 −1 0 −2 −2 1 1 1 0 0 0 0 −ζ12−1−ζ12 ζ12−1+ζ12 0 0 −ζ12−1−ζ12 −ζ12−1−ζ12 ζ12−1+ζ12 ζ12−1+ζ12
72.28.2e2 R 2 −2 0 0 2 2 −1 −1 −1 0 −2 −2 1 1 1 0 0 0 0 ζ12−1+ζ12 −ζ12−1−ζ12 0 0 ζ12−1+ζ12 ζ12−1+ζ12 −ζ12−1−ζ12 −ζ12−1−ζ12
72.28.2f1 C 2 −2 0 0 2ζ3−1 2ζ3 2 2ζ3−1 2ζ3 0 −2ζ3−1 −2ζ3 −2 −2ζ3 −2ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0
72.28.2f2 C 2 −2 0 0 2ζ3 2ζ3−1 2 2ζ3 2ζ3−1 0 −2ζ3 −2ζ3−1 −2 −2ζ3−1 −2ζ3 0 0 0 0 0 0 0 0 0 0 0 0
72.28.2g1 C 2 2 0 0 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 −2 2ζ3−1 2ζ3 −1 −ζ3 −ζ3−1 0 0 0 0 1 1 −2ζ3 −2ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
72.28.2g2 C 2 2 0 0 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 −2 2ζ3 2ζ3−1 −1 −ζ3−1 −ζ3 0 0 0 0 1 1 −2ζ3−1 −2ζ3 ζ3 ζ3−1 ζ3−1 ζ3
72.28.2h1 C 2 −2 0 0 −2ζ122 2ζ124 −1 ζ122 −ζ124 0 2ζ122 −2ζ124 1 ζ124 −ζ122 0 0 0 0 −ζ12−1−ζ12 ζ12−1+ζ12 0 0 ζ12+ζ123 ζ12−2ζ123 −ζ12+2ζ123 −ζ12−ζ123
72.28.2h2 C 2 −2 0 0 2ζ124 −2ζ122 −1 −ζ124 ζ122 0 −2ζ124 2ζ122 1 −ζ122 ζ124 0 0 0 0 −ζ12−1−ζ12 ζ12−1+ζ12 0 0 ζ12−2ζ123 ζ12+ζ123 −ζ12−ζ123 −ζ12+2ζ123
72.28.2h3 C 2 −2 0 0 −2ζ122 2ζ124 −1 ζ122 −ζ124 0 2ζ122 −2ζ124 1 ζ124 −ζ122 0 0 0 0 ζ12−1+ζ12 −ζ12−1−ζ12 0 0 −ζ12−ζ123 −ζ12+2ζ123 ζ12−2ζ123 ζ12+ζ123
72.28.2h4 C 2 −2 0 0 2ζ124 −2ζ122 −1 −ζ124 ζ122 0 −2ζ124 2ζ122 1 −ζ122 ζ124 0 0 0 0 ζ12−1+ζ12 −ζ12−1−ζ12 0 0 −ζ12+2ζ123 −ζ12−ζ123 ζ12+ζ123 ζ12−2ζ123

magma: CharacterTable(G);