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Properties

Label 36T48
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times D_{18}$

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Show commands: Magma

magma: G := TransitiveGroup(36, 48);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $48$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times D_{18}$
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:  (1,4)(2,3)(5,36)(6,35)(7,34)(8,33)(9,29)(10,30)(11,32)(12,31)(13,26)(14,25)(15,27)(16,28)(17,23)(18,24)(19,21)(20,22), (1,29,24,13,5,35,25,17,10,3,31,22,16,7,33,27,19,11)(2,30,23,14,6,36,26,18,9,4,32,21,15,8,34,28,20,12), (1,19)(2,20)(3,17)(4,18)(5,16)(6,15)(7,13)(8,14)(21,36)(22,35)(23,34)(24,33)(25,31)(26,32)(27,29)(28,30)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$
$8$:  $C_2^3$
$12$:  $D_{6}$ x 3
$18$:  $D_{9}$
$24$:  $S_3 \times C_2^2$
$36$:  $D_{18}$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $D_{6}$ x 3

Degree 9: $D_{9}$

Degree 12: $S_3 \times C_2^2$

Degree 18: $D_{18}$ x 3

Low degree siblings

36T48 x 3

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 $2^{18}$ $1$ $2$ $18$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)$
2B $2^{18}$ $1$ $2$ $18$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)$
2C $2^{18}$ $1$ $2$ $18$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
2D $2^{18}$ $9$ $2$ $18$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,36)(10,35)(11,33)(12,34)(13,31)(14,32)(15,30)(16,29)(17,25)(18,26)(19,27)(20,28)(21,23)(22,24)$
2E $2^{18}$ $9$ $2$ $18$ $( 1, 4)( 2, 3)( 5,36)( 6,35)( 7,34)( 8,33)( 9,29)(10,30)(11,32)(12,31)(13,26)(14,25)(15,27)(16,28)(17,23)(18,24)(19,21)(20,22)$
2F $2^{16},1^{4}$ $9$ $2$ $16$ $( 5,33)( 6,34)( 7,35)( 8,36)( 9,32)(10,31)(11,29)(12,30)(13,27)(14,28)(15,26)(16,25)(17,22)(18,21)(19,24)(20,23)$
2G $2^{18}$ $9$ $2$ $18$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,33)(10,34)(11,36)(12,35)(13,30)(14,29)(15,31)(16,32)(17,28)(18,27)(19,26)(20,25)(21,22)(23,24)$
3A $3^{12}$ $2$ $3$ $24$ $( 1,16,25)( 2,15,26)( 3,13,27)( 4,14,28)( 5,19,31)( 6,20,32)( 7,17,29)( 8,18,30)( 9,23,34)(10,24,33)(11,22,35)(12,21,36)$
6A $6^{6}$ $2$ $6$ $30$ $( 1,26,16, 2,25,15)( 3,28,13, 4,27,14)( 5,32,19, 6,31,20)( 7,30,17, 8,29,18)( 9,33,23,10,34,24)(11,36,22,12,35,21)$
6B $6^{6}$ $2$ $6$ $30$ $( 1,27,16, 3,25,13)( 2,28,15, 4,26,14)( 5,29,19, 7,31,17)( 6,30,20, 8,32,18)( 9,36,23,12,34,21)(10,35,24,11,33,22)$
6C $6^{6}$ $2$ $6$ $30$ $( 1,14,25, 4,16,28)( 2,13,26, 3,15,27)( 5,18,31, 8,19,30)( 6,17,32, 7,20,29)( 9,22,34,11,23,35)(10,21,33,12,24,36)$
9A1 $9^{4}$ $2$ $9$ $32$ $( 1,31,24,16, 5,33,25,19,10)( 2,32,23,15, 6,34,26,20, 9)( 3,29,22,13, 7,35,27,17,11)( 4,30,21,14, 8,36,28,18,12)$
9A2 $9^{4}$ $2$ $9$ $32$ $( 1,19,33,16,31,10,25, 5,24)( 2,20,34,15,32, 9,26, 6,23)( 3,17,35,13,29,11,27, 7,22)( 4,18,36,14,30,12,28, 8,21)$
9A4 $9^{4}$ $2$ $9$ $32$ $( 1, 5,10,16,19,24,25,31,33)( 2, 6, 9,15,20,23,26,32,34)( 3, 7,11,13,17,22,27,29,35)( 4, 8,12,14,18,21,28,30,36)$
18A1 $18^{2}$ $2$ $18$ $34$ $( 1,23, 5,26,10,32,16,34,19, 2,24, 6,25, 9,31,15,33,20)( 3,21, 7,28,11,30,13,36,17, 4,22, 8,27,12,29,14,35,18)$
18A5 $18^{2}$ $2$ $18$ $34$ $( 1,35,31,27,24,17,16,11, 5, 3,33,29,25,22,19,13,10, 7)( 2,36,32,28,23,18,15,12, 6, 4,34,30,26,21,20,14, 9, 8)$
18A7 $18^{2}$ $2$ $18$ $34$ $( 1,11,19,27,33, 7,16,22,31, 3,10,17,25,35, 5,13,24,29)( 2,12,20,28,34, 8,15,21,32, 4, 9,18,26,36, 6,14,23,30)$
18B1 $18^{2}$ $2$ $18$ $34$ $( 1,30,24,14, 5,36,25,18,10, 4,31,21,16, 8,33,28,19,12)( 2,29,23,13, 6,35,26,17, 9, 3,32,22,15, 7,34,27,20,11)$
18B5 $18^{2}$ $2$ $18$ $34$ $( 1,22, 5,27,10,29,16,35,19, 3,24, 7,25,11,31,13,33,17)( 2,21, 6,28, 9,30,15,36,20, 4,23, 8,26,12,32,14,34,18)$
18B7 $18^{2}$ $2$ $18$ $34$ $( 1, 8,10,14,19,21,25,30,33, 4, 5,12,16,18,24,28,31,36)( 2, 7, 9,13,20,22,26,29,34, 3, 6,11,15,17,23,27,32,35)$
18C1 $18^{2}$ $2$ $18$ $34$ $( 1,34,31,26,24,20,16, 9, 5, 2,33,32,25,23,19,15,10, 6)( 3,36,29,28,22,18,13,12, 7, 4,35,30,27,21,17,14,11, 8)$
18C5 $18^{2}$ $2$ $18$ $34$ $( 1,18,33,14,31,12,25, 8,24, 4,19,36,16,30,10,28, 5,21)( 2,17,34,13,32,11,26, 7,23, 3,20,35,15,29, 9,27, 6,22)$
18C7 $18^{2}$ $2$ $18$ $34$ $( 1, 9,19,26,33, 6,16,23,31, 2,10,20,25,34, 5,15,24,32)( 3,12,17,28,35, 8,13,21,29, 4,11,18,27,36, 7,14,22,30)$

Malle's constant $a(G)$:     $1/16$

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:  72.17
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 3A 6A 6B 6C 9A1 9A2 9A4 18A1 18A5 18A7 18B1 18B5 18B7 18C1 18C5 18C7
Size 1 1 1 1 9 9 9 9 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A 3A 3A 3A 9A2 9A4 9A1 9A4 9A1 9A2 9A2 9A4 9A1 9A1 9A4 9A2
3 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 2C 2A 2B 3A 3A 3A 6C 6A 6A 6B 6A 6B 6C 6B 6C
Type
72.17.1a 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
72.17.1b 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
72.17.1c 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
72.17.1d 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
72.17.1e 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
72.17.1f 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
72.17.1g 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
72.17.1h 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
72.17.2a R 2 2 2 2 0 0 0 0 2 2 2 2 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1
72.17.2b R 2 −2 −2 2 0 0 0 0 2 −2 −2 2 −1 −1 −1 1 1 1 1 1 1 −1 −1 −1
72.17.2c R 2 −2 2 −2 0 0 0 0 2 −2 2 −2 −1 −1 −1 1 1 1 −1 −1 −1 1 1 1
72.17.2d R 2 2 −2 −2 0 0 0 0 2 2 −2 −2 −1 −1 −1 −1 −1 −1 1 1 1 1 1 1
72.17.2e1 R 2 2 2 2 0 0 0 0 −1 −1 −1 −1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94
72.17.2e2 R 2 2 2 2 0 0 0 0 −1 −1 −1 −1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92
72.17.2e3 R 2 2 2 2 0 0 0 0 −1 −1 −1 −1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9
72.17.2f1 R 2 −2 −2 2 0 0 0 0 −1 1 1 −1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94
72.17.2f2 R 2 −2 −2 2 0 0 0 0 −1 1 1 −1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92
72.17.2f3 R 2 −2 −2 2 0 0 0 0 −1 1 1 −1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9
72.17.2g1 R 2 −2 2 −2 0 0 0 0 −1 1 −1 1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94
72.17.2g2 R 2 −2 2 −2 0 0 0 0 −1 1 −1 1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92
72.17.2g3 R 2 −2 2 −2 0 0 0 0 −1 1 −1 1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9
72.17.2h1 R 2 2 −2 −2 0 0 0 0 −1 −1 1 1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94
72.17.2h2 R 2 2 −2 −2 0 0 0 0 −1 −1 1 1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92
72.17.2h3 R 2 2 −2 −2 0 0 0 0 −1 −1 1 1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9

magma: CharacterTable(G);