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Properties

Label 36T45
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_4\times D_9$

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Show commands: Magma

magma: G := TransitiveGroup(36, 45);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $45$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4\times D_9$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
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)(21,33,22,34)(23,35,24,36)(25,29,26,30)(27,31,28,32), (1,23)(2,24)(3,22)(4,21)(5,20)(6,19)(7,17)(8,18)(9,16)(10,15)(11,13)(12,14)(25,33)(26,34)(27,35)(28,36)(29,30)(31,32)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$
$18$:  $D_{9}$
$24$:  $S_3 \times C_4$
$36$:  $D_{18}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$

Degree 6: $D_{6}$

Degree 9: $D_{9}$

Degree 12: $S_3 \times C_4$

Degree 18: $D_{18}$

Low degree siblings

36T45

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

Malle's constant $a(G)$:     $1/16$

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:  72.5
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 6A 9A1 9A2 9A4 12A1 12A-1 18A1 18A5 18A7 36A1 36A-1 36A5 36A-5 36A7 36A-7
Size 1 1 9 9 2 1 1 9 9 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 3A 2A 2A 2A 2A 3A 9A2 9A4 9A1 6A 6A 9A1 9A4 9A2 18A1 18A7 18A5 18A1 18A5 18A7
3 P 1A 2A 2B 2C 1A 4A-1 4A1 4B-1 4B1 2A 3A 3A 3A 4A1 4A-1 6A 6A 6A 12A1 12A-1 12A1 12A-1 12A-1 12A1
Type
72.5.1a 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
72.5.1b 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
72.5.1c 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
72.5.1d 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
72.5.1e1 C 1 −1 −1 1 1 −i i i −i −1 1 1 1 i −i −1 −1 −1 −i i −i i i −i
72.5.1e2 C 1 −1 −1 1 1 i −i −i i −1 1 1 1 −i i −1 −1 −1 i −i i −i −i i
72.5.1f1 C 1 −1 1 −1 1 −i i −i i −1 1 1 1 i −i −1 −1 −1 −i i −i i i −i
72.5.1f2 C 1 −1 1 −1 1 i −i i −i −1 1 1 1 −i i −1 −1 −1 i −i i −i −i i
72.5.2a R 2 2 0 0 2 2 2 0 0 2 −1 −1 −1 2 2 −1 −1 −1 −1 −1 −1 −1 −1 −1
72.5.2b R 2 2 0 0 2 −2 −2 0 0 2 −1 −1 −1 −2 −2 −1 −1 −1 1 1 1 1 1 1
72.5.2c1 C 2 −2 0 0 2 −2i 2i 0 0 −2 −1 −1 −1 2i −2i 1 1 1 i −i i −i −i i
72.5.2c2 C 2 −2 0 0 2 2i −2i 0 0 −2 −1 −1 −1 −2i 2i 1 1 1 −i i −i i i −i
72.5.2d1 R 2 2 0 0 −1 2 2 0 0 −1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 −1 −1 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−2+ζ92
72.5.2d2 R 2 2 0 0 −1 2 2 0 0 −1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 −1 −1 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−1+ζ9
72.5.2d3 R 2 2 0 0 −1 2 2 0 0 −1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 −1 −1 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−4+ζ94
72.5.2e1 R 2 2 0 0 −1 −2 −2 0 0 −1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 1 1 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 −ζ9−1−ζ9 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−2−ζ92
72.5.2e2 R 2 2 0 0 −1 −2 −2 0 0 −1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 1 1 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 −ζ9−4−ζ94 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−1−ζ9
72.5.2e3 R 2 2 0 0 −1 −2 −2 0 0 −1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 1 1 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 −ζ9−2−ζ92 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−4−ζ94
72.5.2f1 C 2 −2 0 0 −1 −2ζ369 2ζ369 0 0 1 −ζ36−2−ζ362 ζ36−4+ζ364 ζ36−8+ζ368 −ζ369 ζ369 −ζ36−8−ζ368 −ζ36−4−ζ364 ζ36−2+ζ362 ζ36−ζ365−ζ367 −ζ36+ζ365+ζ367 ζ367+ζ3611 −ζ367−ζ3611 ζ36−ζ365+ζ3611 −ζ36+ζ365−ζ3611
72.5.2f2 C 2 −2 0 0 −1 2ζ369 −2ζ369 0 0 1 −ζ36−2−ζ362 ζ36−4+ζ364 ζ36−8+ζ368 ζ369 −ζ369 −ζ36−8−ζ368 −ζ36−4−ζ364 ζ36−2+ζ362 −ζ36+ζ365+ζ367 ζ36−ζ365−ζ367 −ζ367−ζ3611 ζ367+ζ3611 −ζ36+ζ365−ζ3611 ζ36−ζ365+ζ3611
72.5.2f3 C 2 −2 0 0 −1 −2ζ369 2ζ369 0 0 1 ζ36−8+ζ368 −ζ36−2−ζ362 ζ36−4+ζ364 −ζ369 ζ369 −ζ36−4−ζ364 ζ36−2+ζ362 −ζ36−8−ζ368 ζ367+ζ3611 −ζ367−ζ3611 −ζ36+ζ365−ζ3611 ζ36−ζ365+ζ3611 −ζ36+ζ365+ζ367 ζ36−ζ365−ζ367
72.5.2f4 C 2 −2 0 0 −1 2ζ369 −2ζ369 0 0 1 ζ36−8+ζ368 −ζ36−2−ζ362 ζ36−4+ζ364 ζ369 −ζ369 −ζ36−4−ζ364 ζ36−2+ζ362 −ζ36−8−ζ368 −ζ367−ζ3611 ζ367+ζ3611 ζ36−ζ365+ζ3611 −ζ36+ζ365−ζ3611 ζ36−ζ365−ζ367 −ζ36+ζ365+ζ367
72.5.2f5 C 2 −2 0 0 −1 −2ζ369 2ζ369 0 0 1 ζ36−4+ζ364 ζ36−8+ζ368 −ζ36−2−ζ362 −ζ369 ζ369 ζ36−2+ζ362 −ζ36−8−ζ368 −ζ36−4−ζ364 −ζ36+ζ365−ζ3611 ζ36−ζ365+ζ3611 ζ36−ζ365−ζ367 −ζ36+ζ365+ζ367 −ζ367−ζ3611 ζ367+ζ3611
72.5.2f6 C 2 −2 0 0 −1 2ζ369 −2ζ369 0 0 1 ζ36−4+ζ364 ζ36−8+ζ368 −ζ36−2−ζ362 ζ369 −ζ369 ζ36−2+ζ362 −ζ36−8−ζ368 −ζ36−4−ζ364 ζ36−ζ365+ζ3611 −ζ36+ζ365−ζ3611 −ζ36+ζ365+ζ367 ζ36−ζ365−ζ367 ζ367+ζ3611 −ζ367−ζ3611

magma: CharacterTable(G);