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Properties

Label 36T25
Degree $36$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^2:D_9$

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Show commands: Magma

magma: G := TransitiveGroup(36, 25);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $25$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^2:D_9$
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:  (1,22,2,21)(3,23)(4,24)(5,20,6,19)(7,18,8,17)(9,16,10,15)(11,14)(12,13)(25,34)(26,33)(27,35,28,36)(29,31,30,32), (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)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$18$:  $D_{9}$
$24$:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_3$, $S_4$, $S_4$

Degree 9: $D_{9}$

Degree 12: $S_4$

Degree 18: $D_9$, 18T38, 18T39

Low degree siblings

18T38, 18T39, 36T57

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

Malle's constant $a(G)$:     $1/12$

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:  72.15
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A 4A 6A 9A1 9A2 9A4
Size 1 3 18 2 18 6 8 8 8
2 P 1A 1A 1A 3A 2A 3A 9A1 9A2 9A4
3 P 1A 2A 2B 1A 4A 2A 3A 3A 3A
Type
72.15.1a R 1 1 1 1 1 1 1 1 1
72.15.1b R 1 1 −1 1 −1 1 1 1 1
72.15.2a R 2 2 0 2 0 2 −1 −1 −1
72.15.2b1 R 2 2 0 −1 0 −1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92
72.15.2b2 R 2 2 0 −1 0 −1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9
72.15.2b3 R 2 2 0 −1 0 −1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94
72.15.3a R 3 −1 1 3 −1 −1 0 0 0
72.15.3b R 3 −1 −1 3 1 −1 0 0 0
72.15.6a R 6 −2 0 −3 0 1 0 0 0

magma: CharacterTable(G);