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Properties

Label 36T46
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_9:D_4$

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Show commands: Magma

magma: G := TransitiveGroup(36, 46);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $46$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_9:D_4$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,18,2,17)(3,19,4,20)(5,14,6,13)(7,16,8,15)(9,11,10,12)(21,34,22,33)(23,35,24,36)(25,30,26,29)(27,31,28,32), (1,16)(2,15)(3,14)(4,13)(5,10)(6,9)(7,12)(8,11)(17,36)(18,35)(19,33)(20,34)(21,29)(22,30)(23,32)(24,31)(27,28)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$8$:  $D_{4}$
$12$:  $D_{6}$
$18$:  $D_{9}$
$24$:  $(C_6\times C_2):C_2$
$36$:  $D_{18}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $D_{6}$

Degree 9: $D_{9}$

Degree 12: $(C_6\times C_2):C_2$

Degree 18: $D_{18}$

Low degree siblings

36T24

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 $2^{18}$ $2$ $2$ $18$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)$
2C $2^{17},1^{2}$ $18$ $2$ $17$ $( 1,32)( 2,31)( 3,29)( 4,30)( 5,26)( 6,25)( 7,27)( 8,28)( 9,24)(10,23)(11,22)(12,21)(13,17)(14,18)(15,19)(16,20)(33,34)$
3A $3^{12}$ $2$ $3$ $24$ $( 1,16,25)( 2,15,26)( 3,13,27)( 4,14,28)( 5,19,31)( 6,20,32)( 7,17,29)( 8,18,30)( 9,23,34)(10,24,33)(11,22,35)(12,21,36)$
4A $4^{9}$ $18$ $4$ $27$ $( 1,29, 2,30)( 3,32, 4,31)( 5,27, 6,28)( 7,26, 8,25)( 9,21,10,22)(11,23,12,24)(13,20,14,19)(15,18,16,17)(33,35,34,36)$
6A $6^{6}$ $2$ $6$ $30$ $( 1,26,16, 2,25,15)( 3,28,13, 4,27,14)( 5,32,19, 6,31,20)( 7,30,17, 8,29,18)( 9,33,23,10,34,24)(11,36,22,12,35,21)$
6B1 $6^{6}$ $2$ $6$ $30$ $( 1,28,16, 4,25,14)( 2,27,15, 3,26,13)( 5,30,19, 8,31,18)( 6,29,20, 7,32,17)( 9,35,23,11,34,22)(10,36,24,12,33,21)$
6B-1 $6^{6}$ $2$ $6$ $30$ $( 1,14,25, 4,16,28)( 2,13,26, 3,15,27)( 5,18,31, 8,19,30)( 6,17,32, 7,20,29)( 9,22,34,11,23,35)(10,21,33,12,24,36)$
9A1 $9^{4}$ $2$ $9$ $32$ $( 1, 5,10,16,19,24,25,31,33)( 2, 6, 9,15,20,23,26,32,34)( 3, 7,11,13,17,22,27,29,35)( 4, 8,12,14,18,21,28,30,36)$
9A2 $9^{4}$ $2$ $9$ $32$ $( 1,31,24,16, 5,33,25,19,10)( 2,32,23,15, 6,34,26,20, 9)( 3,29,22,13, 7,35,27,17,11)( 4,30,21,14, 8,36,28,18,12)$
9A4 $9^{4}$ $2$ $9$ $32$ $( 1,24, 5,25,10,31,16,33,19)( 2,23, 6,26, 9,32,15,34,20)( 3,22, 7,27,11,29,13,35,17)( 4,21, 8,28,12,30,14,36,18)$
18A1 $18^{2}$ $2$ $18$ $34$ $( 1,18,33,14,31,12,25, 8,24, 4,19,36,16,30,10,28, 5,21)( 2,17,34,13,32,11,26, 7,23, 3,20,35,15,29, 9,27, 6,22)$
18A5 $18^{2}$ $2$ $18$ $34$ $( 1,12,19,28,33, 8,16,21,31, 4,10,18,25,36, 5,14,24,30)( 2,11,20,27,34, 7,15,22,32, 3, 9,17,26,35, 6,13,23,29)$
18A7 $18^{2}$ $2$ $18$ $34$ $( 1,21, 5,28,10,30,16,36,19, 4,24, 8,25,12,31,14,33,18)( 2,22, 6,27, 9,29,15,35,20, 3,23, 7,26,11,32,13,34,17)$
18B1 $18^{2}$ $2$ $18$ $34$ $( 1,20,33,15,31, 9,25, 6,24, 2,19,34,16,32,10,26, 5,23)( 3,18,35,14,29,12,27, 8,22, 4,17,36,13,30,11,28, 7,21)$
18B-1 $18^{2}$ $2$ $18$ $34$ $( 1,34,31,26,24,20,16, 9, 5, 2,33,32,25,23,19,15,10, 6)( 3,36,29,28,22,18,13,12, 7, 4,35,30,27,21,17,14,11, 8)$
18B5 $18^{2}$ $2$ $18$ $34$ $( 1, 8,10,14,19,21,25,30,33, 4, 5,12,16,18,24,28,31,36)( 2, 7, 9,13,20,22,26,29,34, 3, 6,11,15,17,23,27,32,35)$
18B-5 $18^{2}$ $2$ $18$ $34$ $( 1,30,24,14, 5,36,25,18,10, 4,31,21,16, 8,33,28,19,12)( 2,29,23,13, 6,35,26,17, 9, 3,32,22,15, 7,34,27,20,11)$
18B7 $18^{2}$ $2$ $18$ $34$ $( 1,36,31,28,24,18,16,12, 5, 4,33,30,25,21,19,14,10, 8)( 2,35,32,27,23,17,15,11, 6, 3,34,29,26,22,20,13, 9, 7)$
18B-7 $18^{2}$ $2$ $18$ $34$ $( 1, 9,19,26,33, 6,16,23,31, 2,10,20,25,34, 5,15,24,32)( 3,12,17,28,35, 8,13,21,29, 4,11,18,27,36, 7,14,22,30)$

Malle's constant $a(G)$:     $1/17$

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:  72.8
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 4A 6A 6B1 6B-1 9A1 9A2 9A4 18A1 18A5 18A7 18B1 18B-1 18B5 18B-5 18B7 18B-7
Size 1 1 2 18 2 18 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 3A 2A 3A 3A 3A 9A1 9A2 9A4 9A4 9A2 9A4 9A4 9A1 9A1 9A2 9A1 9A2
3 P 1A 2A 2B 2C 1A 4A 2A 2B 2B 3A 3A 3A 6B-1 6B1 6B1 6A 6A 6B-1 6B-1 6B1 6A
Type
72.8.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.8.1b R 1 1 −1 −1 1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
72.8.1c R 1 1 −1 1 1 −1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
72.8.1d R 1 1 1 −1 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.8.2a R 2 2 2 0 2 0 2 2 2 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1
72.8.2b R 2 −2 0 0 2 0 −2 0 0 2 2 2 −2 −2 −2 0 0 0 0 0 0
72.8.2c R 2 2 −2 0 2 0 2 −2 −2 −1 −1 −1 −1 −1 −1 1 1 1 1 1 1
72.8.2d1 C 2 −2 0 0 2 0 −2 0 0 −1 −1 −1 1 1 1 −1−2ζ3 1+2ζ3 1+2ζ3 −1−2ζ3 −1−2ζ3 1+2ζ3
72.8.2d2 C 2 −2 0 0 2 0 −2 0 0 −1 −1 −1 1 1 1 1+2ζ3 −1−2ζ3 −1−2ζ3 1+2ζ3 1+2ζ3 −1−2ζ3
72.8.2e1 R 2 2 2 0 −1 0 −1 −1 −1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−4+ζ94
72.8.2e2 R 2 2 2 0 −1 0 −1 −1 −1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−2+ζ92
72.8.2e3 R 2 2 2 0 −1 0 −1 −1 −1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−1+ζ9
72.8.2f1 R 2 2 −2 0 −1 0 −1 1 1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 −ζ9−2−ζ92 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−4−ζ94
72.8.2f2 R 2 2 −2 0 −1 0 −1 1 1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 −ζ9−1−ζ9 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−2−ζ92
72.8.2f3 R 2 2 −2 0 −1 0 −1 1 1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 −ζ9−4−ζ94 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−1−ζ9
72.8.2g1 C 2 −2 0 0 −1 0 1 −1−2ζ93 1+2ζ93 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92 ζ9+ζ92+ζ94 −ζ9−ζ92−ζ94 ζ9−4+ζ9+ζ92 −ζ9−4−ζ9−ζ92 ζ9−4−ζ94 −ζ9−4+ζ94
72.8.2g2 C 2 −2 0 0 −1 0 1 1+2ζ93 −1−2ζ93 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−ζ92−ζ94 ζ9+ζ92+ζ94 −ζ9−4−ζ9−ζ92 ζ9−4+ζ9+ζ92 −ζ9−4+ζ94 ζ9−4−ζ94
72.8.2g3 C 2 −2 0 0 −1 0 1 −1−2ζ93 1+2ζ93 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−4−ζ9−ζ92 ζ9−4+ζ9+ζ92 −ζ9−4+ζ94 ζ9−4−ζ94 ζ9+ζ92+ζ94 −ζ9−ζ92−ζ94
72.8.2g4 C 2 −2 0 0 −1 0 1 1+2ζ93 −1−2ζ93 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9 ζ9−4+ζ9+ζ92 −ζ9−4−ζ9−ζ92 ζ9−4−ζ94 −ζ9−4+ζ94 −ζ9−ζ92−ζ94 ζ9+ζ92+ζ94
72.8.2g5 C 2 −2 0 0 −1 0 1 −1−2ζ93 1+2ζ93 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94 ζ9−4−ζ94 −ζ9−4+ζ94 −ζ9−ζ92−ζ94 ζ9+ζ92+ζ94 −ζ9−4−ζ9−ζ92 ζ9−4+ζ9+ζ92
72.8.2g6 C 2 −2 0 0 −1 0 1 1+2ζ93 −1−2ζ93 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−4+ζ94 ζ9−4−ζ94 ζ9+ζ92+ζ94 −ζ9−ζ92−ζ94 ζ9−4+ζ9+ζ92 −ζ9−4−ζ9−ζ92

magma: CharacterTable(G);