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Properties

Label 36T16
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^2:C_{18}$

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Show commands: Magma

magma: G := TransitiveGroup(36, 16);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $16$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^2:C_{18}$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $12$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,35,32,27,21,20,13,11,5)(2,36,31,28,22,19,14,12,6)(3,34,30,25,24,18,16,10,7)(4,33,29,26,23,17,15,9,8), (1,16,27,3,13,25)(2,15,28,4,14,26)(5,17,32,8,20,29)(6,18,31,7,19,30)(9,22,33,12,23,36)(10,21,34,11,24,35)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$9$:  $C_9$
$12$:  $A_4$
$18$:  $C_{18}$
$24$:  $A_4\times C_2$
$36$:  $C_2^2 : C_9$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: None

Degree 6: $C_6$, $A_4$, $A_4\times C_2$

Degree 9: $C_9$

Degree 12: $A_4 \times C_2$

Degree 18: $C_{18}$, $C_2^2 : C_9$, 18T26

Low degree siblings

18T26, 36T30

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

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:  72.16
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 9A1 9A-1 9A2 9A-2 9A4 9A-4 18A1 18A-1 18A5 18A-5 18A7 18A-7
Size 1 1 3 3 1 1 1 1 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4
2 P 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 3A-1 3A1 3A-1 3A1 9A1 9A-2 9A-4 9A2 9A4 9A-1 9A4 9A-1 9A-4 9A-2 9A1 9A2
3 P 1A 2A 2B 2C 1A 1A 2A 2A 2B 2C 2C 2B 3A-1 3A-1 3A1 3A1 3A-1 3A1 6A1 6A-1 6A-1 6A1 6A1 6A-1
Type
72.16.1a 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
72.16.1b 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
72.16.1c1 C 1 1 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
72.16.1c2 C 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
72.16.1d1 C 1 −1 1 −1 1 1 −1 −1 1 1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
72.16.1d2 C 1 −1 1 −1 1 1 −1 −1 1 1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
72.16.1e1 C 1 1 1 1 ζ9−3 ζ93 ζ93 ζ9−3 ζ93 ζ9−3 ζ9−3 ζ93 ζ9−4 ζ94 ζ9 ζ9−1 ζ92 ζ9−2 ζ9−2 ζ92 ζ9−1 ζ9 ζ94 ζ9−4
72.16.1e2 C 1 1 1 1 ζ93 ζ9−3 ζ9−3 ζ93 ζ9−3 ζ93 ζ93 ζ9−3 ζ94 ζ9−4 ζ9−1 ζ9 ζ9−2 ζ92 ζ92 ζ9−2 ζ9 ζ9−1 ζ9−4 ζ94
72.16.1e3 C 1 1 1 1 ζ9−3 ζ93 ζ93 ζ9−3 ζ93 ζ9−3 ζ9−3 ζ93 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−1 ζ9 ζ9 ζ9−1 ζ9−4 ζ94 ζ9−2 ζ92
72.16.1e4 C 1 1 1 1 ζ93 ζ9−3 ζ9−3 ζ93 ζ9−3 ζ93 ζ93 ζ9−3 ζ9−2 ζ92 ζ9−4 ζ94 ζ9 ζ9−1 ζ9−1 ζ9 ζ94 ζ9−4 ζ92 ζ9−2
72.16.1e5 C 1 1 1 1 ζ9−3 ζ93 ζ93 ζ9−3 ζ93 ζ9−3 ζ9−3 ζ93 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−4 ζ94 ζ94 ζ9−4 ζ92 ζ9−2 ζ9 ζ9−1
72.16.1e6 C 1 1 1 1 ζ93 ζ9−3 ζ9−3 ζ93 ζ9−3 ζ93 ζ93 ζ9−3 ζ9 ζ9−1 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−4 ζ94 ζ9−2 ζ92 ζ9−1 ζ9
72.16.1f1 C 1 −1 1 −1 ζ9−3 ζ93 −ζ93 −ζ9−3 ζ93 ζ9−3 −ζ9−3 −ζ93 ζ9−4 ζ94 ζ9 ζ9−1 ζ92 ζ9−2 −ζ9−2 −ζ92 −ζ9−1 −ζ9 −ζ94 −ζ9−4
72.16.1f2 C 1 −1 1 −1 ζ93 ζ9−3 −ζ9−3 −ζ93 ζ9−3 ζ93 −ζ93 −ζ9−3 ζ94 ζ9−4 ζ9−1 ζ9 ζ9−2 ζ92 −ζ92 −ζ9−2 −ζ9 −ζ9−1 −ζ9−4 −ζ94
72.16.1f3 C 1 −1 1 −1 ζ9−3 ζ93 −ζ93 −ζ9−3 ζ93 ζ9−3 −ζ9−3 −ζ93 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−1 ζ9 −ζ9 −ζ9−1 −ζ9−4 −ζ94 −ζ9−2 −ζ92
72.16.1f4 C 1 −1 1 −1 ζ93 ζ9−3 −ζ9−3 −ζ93 ζ9−3 ζ93 −ζ93 −ζ9−3 ζ9−2 ζ92 ζ9−4 ζ94 ζ9 ζ9−1 −ζ9−1 −ζ9 −ζ94 −ζ9−4 −ζ92 −ζ9−2
72.16.1f5 C 1 −1 1 −1 ζ9−3 ζ93 −ζ93 −ζ9−3 ζ93 ζ9−3 −ζ9−3 −ζ93 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−4 ζ94 −ζ94 −ζ9−4 −ζ92 −ζ9−2 −ζ9 −ζ9−1
72.16.1f6 C 1 −1 1 −1 ζ93 ζ9−3 −ζ9−3 −ζ93 ζ9−3 ζ93 −ζ93 −ζ9−3 ζ9 ζ9−1 ζ92 ζ9−2 ζ94 ζ9−4 −ζ9−4 −ζ94 −ζ9−2 −ζ92 −ζ9−1 −ζ9
72.16.3a R 3 3 −1 −1 3 3 3 3 −1 −1 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0
72.16.3b R 3 −3 −1 1 3 3 −3 −3 −1 −1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
72.16.3c1 C 3 3 −1 −1 3ζ3−1 3ζ3 3ζ3 3ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 0 0 0 0 0 0 0 0 0 0 0 0
72.16.3c2 C 3 3 −1 −1 3ζ3 3ζ3−1 3ζ3−1 3ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0
72.16.3d1 C 3 −3 −1 1 3ζ3−1 3ζ3 −3ζ3 −3ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 0 0 0 0 0 0 0 0 0 0 0 0
72.16.3d2 C 3 −3 −1 1 3ζ3 3ζ3−1 −3ζ3−1 −3ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);