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Properties

Label 36T47
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{36}$

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Show commands: Magma

magma: G := TransitiveGroup(36, 47);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $47$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{36}$
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:  (1,7)(2,8)(3,6)(4,5)(9,36)(10,35)(11,33)(12,34)(13,31)(14,32)(15,30)(16,29)(17,25)(18,26)(19,27)(20,28)(21,23)(22,24), (1,36,31,28,24,18,16,12,5,4,34,29,26,22,20,13,9,7,2,35,32,27,23,17,15,11,6,3,33,30,25,21,19,14,10,8)
magma: Generators(G);
 

Low degree resolvents

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

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: $D_{12}$

Degree 18: $D_{18}$

Low degree siblings

36T47

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 $2^{17},1^{2}$ $18$ $2$ $17$ $( 1,32)( 2,31)( 3,30)( 4,29)( 5,26)( 6,25)( 7,28)( 8,27)( 9,24)(10,23)(11,21)(12,22)(13,18)(14,17)(15,19)(16,20)(35,36)$
2C $2^{18}$ $18$ $2$ $18$ $( 1,27)( 2,28)( 3,25)( 4,26)( 5,22)( 6,21)( 7,24)( 8,23)( 9,18)(10,17)(11,19)(12,20)(13,16)(14,15)(29,34)(30,33)(31,35)(32,36)$
3A $3^{12}$ $2$ $3$ $24$ $( 1,26,15)( 2,25,16)( 3,28,13)( 4,27,14)( 5,32,19)( 6,31,20)( 7,30,18)( 8,29,17)( 9,33,24)(10,34,23)(11,36,22)(12,35,21)$
4A $4^{9}$ $2$ $4$ $27$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)(17,19,18,20)(21,24,22,23)(25,28,26,27)(29,32,30,31)(33,36,34,35)$
6A $6^{6}$ $2$ $6$ $30$ $( 1,16,26, 2,15,25)( 3,14,28, 4,13,27)( 5,20,32, 6,19,31)( 7,17,30, 8,18,29)( 9,23,33,10,24,34)(11,21,36,12,22,35)$
9A1 $9^{4}$ $2$ $9$ $32$ $( 1,24, 5,26, 9,32,15,33,19)( 2,23, 6,25,10,31,16,34,20)( 3,21, 8,28,12,29,13,35,17)( 4,22, 7,27,11,30,14,36,18)$
9A2 $9^{4}$ $2$ $9$ $32$ $( 1,33,32,26,24,19,15, 9, 5)( 2,34,31,25,23,20,16,10, 6)( 3,35,29,28,21,17,13,12, 8)( 4,36,30,27,22,18,14,11, 7)$
9A4 $9^{4}$ $2$ $9$ $32$ $( 1, 9,19,26,33, 5,15,24,32)( 2,10,20,25,34, 6,16,23,31)( 3,12,17,28,35, 8,13,21,29)( 4,11,18,27,36, 7,14,22,30)$
12A1 $12^{3}$ $2$ $12$ $33$ $( 1,14,25, 3,15,27, 2,13,26, 4,16,28)( 5,18,31, 8,19,30, 6,17,32, 7,20,29)( 9,22,34,12,24,36,10,21,33,11,23,35)$
12A5 $12^{3}$ $2$ $12$ $33$ $( 1,27,16, 3,26,14, 2,28,15, 4,25,13)( 5,30,20, 8,32,18, 6,29,19, 7,31,17)( 9,36,23,12,33,22,10,35,24,11,34,21)$
18A1 $18^{2}$ $2$ $18$ $34$ $( 1,31,24,16, 5,34,26,20, 9, 2,32,23,15, 6,33,25,19,10)( 3,30,21,14, 8,36,28,18,12, 4,29,22,13, 7,35,27,17,11)$
18A5 $18^{2}$ $2$ $18$ $34$ $( 1, 6, 9,16,19,23,26,31,33, 2, 5,10,15,20,24,25,32,34)( 3, 7,12,14,17,22,28,30,35, 4, 8,11,13,18,21,27,29,36)$
18A7 $18^{2}$ $2$ $18$ $34$ $( 1,20,33,16,32,10,26, 6,24, 2,19,34,15,31, 9,25, 5,23)( 3,18,35,14,29,11,28, 7,21, 4,17,36,13,30,12,27, 8,22)$
36A1 $36$ $2$ $36$ $35$ $( 1,11,20,28,33, 7,16,21,32, 4,10,17,26,36, 6,13,24,30, 2,12,19,27,34, 8,15,22,31, 3, 9,18,25,35, 5,14,23,29)$
36A5 $36$ $2$ $36$ $35$ $( 1, 7,10,13,19,22,25,29,33, 4, 6,12,15,18,23,28,32,36, 2, 8, 9,14,20,21,26,30,34, 3, 5,11,16,17,24,27,31,35)$
36A7 $36$ $2$ $36$ $35$ $( 1,18,34,13,32,11,25, 8,24, 4,20,35,15,30,10,28, 5,22, 2,17,33,14,31,12,26, 7,23, 3,19,36,16,29, 9,27, 6,21)$
36A11 $36$ $2$ $36$ $35$ $( 1,22, 6,28, 9,30,16,35,19, 4,23, 8,26,11,31,13,33,18, 2,21, 5,27,10,29,15,36,20, 3,24, 7,25,12,32,14,34,17)$
36A13 $36$ $2$ $36$ $35$ $( 1,36,31,28,24,18,16,12, 5, 4,34,29,26,22,20,13, 9, 7, 2,35,32,27,23,17,15,11, 6, 3,33,30,25,21,19,14,10, 8)$
36A17 $36$ $2$ $36$ $35$ $( 1,30,23,13, 5,36,25,17, 9, 4,31,21,15, 7,34,28,19,11, 2,29,24,14, 6,35,26,18,10, 3,32,22,16, 8,33,27,20,12)$

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:  72.6
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 4A 6A 9A1 9A2 9A4 12A1 12A5 18A1 18A5 18A7 36A1 36A5 36A7 36A11 36A13 36A17
Size 1 1 18 18 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 3A 2A 3A 9A2 9A4 9A1 6A 6A 9A1 9A4 9A2 18A7 18A1 18A5 18A5 18A1 18A7
3 P 1A 2A 2B 2C 1A 4A 2A 3A 3A 3A 4A 4A 6A 6A 6A 12A1 12A5 12A5 12A1 12A1 12A5
Type
72.6.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.6.1b R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.6.1c R 1 1 −1 1 1 −1 1 1 1 1 −1 −1 1 1 1 −1 −1 −1 −1 −1 −1
72.6.1d R 1 1 1 −1 1 −1 1 1 1 1 −1 −1 1 1 1 −1 −1 −1 −1 −1 −1
72.6.2a R 2 2 0 0 2 2 2 −1 −1 −1 2 2 −1 −1 −1 −1 −1 −1 −1 −1 −1
72.6.2b R 2 −2 0 0 2 0 −2 2 2 2 0 0 −2 −2 −2 0 0 0 0 0 0
72.6.2c R 2 2 0 0 2 −2 2 −1 −1 −1 −2 −2 −1 −1 −1 1 1 1 1 1 1
72.6.2d1 R 2 −2 0 0 2 0 −2 −1 −1 −1 0 0 1 1 1 −ζ12−1−ζ12 ζ12−1+ζ12 ζ12−1+ζ12 −ζ12−1−ζ12 −ζ12−1−ζ12 ζ12−1+ζ12
72.6.2d2 R 2 −2 0 0 2 0 −2 −1 −1 −1 0 0 1 1 1 ζ12−1+ζ12 −ζ12−1−ζ12 −ζ12−1−ζ12 ζ12−1+ζ12 ζ12−1+ζ12 −ζ12−1−ζ12
72.6.2e1 R 2 2 0 0 −1 2 −1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 −1 −1 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9
72.6.2e2 R 2 2 0 0 −1 2 −1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 −1 −1 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94
72.6.2e3 R 2 2 0 0 −1 2 −1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 −1 −1 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92
72.6.2f1 R 2 2 0 0 −1 −2 −1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 1 1 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−2−ζ92 −ζ9−4−ζ94 −ζ9−1−ζ9
72.6.2f2 R 2 2 0 0 −1 −2 −1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 1 1 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−1−ζ9 −ζ9−2−ζ92 −ζ9−4−ζ94
72.6.2f3 R 2 2 0 0 −1 −2 −1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 1 1 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−4−ζ94 −ζ9−1−ζ9 −ζ9−2−ζ92
72.6.2g1 R 2 −2 0 0 −1 0 1 −ζ36−2−ζ362 ζ36−4+ζ364 ζ36−8+ζ368 −ζ36−3−ζ363 ζ36−3+ζ363 −ζ36−8−ζ368 −ζ36−4−ζ364 ζ36−2+ζ362 ζ36−5+ζ365 −ζ36−7−ζ367 ζ36−1+ζ36 −ζ36−1−ζ36 ζ36−7+ζ367 −ζ36−5−ζ365
72.6.2g2 R 2 −2 0 0 −1 0 1 −ζ36−2−ζ362 ζ36−4+ζ364 ζ36−8+ζ368 ζ36−3+ζ363 −ζ36−3−ζ363 −ζ36−8−ζ368 −ζ36−4−ζ364 ζ36−2+ζ362 −ζ36−5−ζ365 ζ36−7+ζ367 −ζ36−1−ζ36 ζ36−1+ζ36 −ζ36−7−ζ367 ζ36−5+ζ365
72.6.2g3 R 2 −2 0 0 −1 0 1 ζ36−8+ζ368 −ζ36−2−ζ362 ζ36−4+ζ364 −ζ36−3−ζ363 ζ36−3+ζ363 −ζ36−4−ζ364 ζ36−2+ζ362 −ζ36−8−ζ368 ζ36−7+ζ367 ζ36−1+ζ36 −ζ36−5−ζ365 ζ36−5+ζ365 −ζ36−1−ζ36 −ζ36−7−ζ367
72.6.2g4 R 2 −2 0 0 −1 0 1 ζ36−8+ζ368 −ζ36−2−ζ362 ζ36−4+ζ364 ζ36−3+ζ363 −ζ36−3−ζ363 −ζ36−4−ζ364 ζ36−2+ζ362 −ζ36−8−ζ368 −ζ36−7−ζ367 −ζ36−1−ζ36 ζ36−5+ζ365 −ζ36−5−ζ365 ζ36−1+ζ36 ζ36−7+ζ367
72.6.2g5 R 2 −2 0 0 −1 0 1 ζ36−4+ζ364 ζ36−8+ζ368 −ζ36−2−ζ362 −ζ36−3−ζ363 ζ36−3+ζ363 ζ36−2+ζ362 −ζ36−8−ζ368 −ζ36−4−ζ364 −ζ36−1−ζ36 −ζ36−5−ζ365 −ζ36−7−ζ367 ζ36−7+ζ367 ζ36−5+ζ365 ζ36−1+ζ36
72.6.2g6 R 2 −2 0 0 −1 0 1 ζ36−4+ζ364 ζ36−8+ζ368 −ζ36−2−ζ362 ζ36−3+ζ363 −ζ36−3−ζ363 ζ36−2+ζ362 −ζ36−8−ζ368 −ζ36−4−ζ364 ζ36−1+ζ36 ζ36−5+ζ365 ζ36−7+ζ367 −ζ36−7−ζ367 −ζ36−5−ζ365 −ζ36−1−ζ36

magma: CharacterTable(G);