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Properties

Label 36T24
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, 24);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $24$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_9:D_4$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $18$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,23,7,27,12,31,16,33,19,2,24,8,28,11,32,15,34,20)(3,22,5,25,10,30,13,35,17)(4,21,6,26,9,29,14,36,18), (1,5,2,6)(3,8,4,7)(9,34,10,33)(11,36,12,35)(13,31,14,32)(15,29,16,30)(17,27,18,28)(19,25,20,26)(21,24,22,23)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$8$:  $D_{4}$
$12$:  $D_{6}$
$18$:  $D_{9}$
$24$:  $(C_6\times C_2):C_2$
$36$:  $D_{18}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $S_3$

Degree 9: $D_{9}$

Degree 12: $(C_6\times C_2):C_2$

Degree 18: $D_9$

Low degree siblings

36T46

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

Malle's constant $a(G)$:     $1/9$

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