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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

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

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 9A2 9A4 9A1 9A4 9A4 9A4 9A1 9A2 9A2 9A1 9A2 9A1
3 P 1A 2A 2B 2C 1A 4A 2A 2B 2B 3A 3A 3A 6A 6B1 6B-1 6A 6B1 6B-1 6B-1 6A 6B1
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);