↑

Properties

Label 12T75
Degree $12$
Order $120$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_2\times A_5$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(12, 75);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $75$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times A_5$
CHM label:   $L(6)[x]2$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,3,5,7,9)(2,4,6,8,12), (1,11)(2,8)(3,9)(10,12), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$60$:  $A_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $\PSL(2,5)$

Low degree siblings

10T11, 12T76, 20T31, 20T36, 24T203, 30T29, 30T30, 40T61

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $15$ $2$ $( 4, 8)( 5, 9)( 6,10)( 7,11)$
$ 5, 5, 1, 1 $ $12$ $5$ $( 2, 4,10, 6, 8)( 3, 5,11, 7, 9)$
$ 5, 5, 1, 1 $ $12$ $5$ $( 2, 6, 4, 8,10)( 3, 7, 5, 9,11)$
$ 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3,12)( 4, 5)( 6,11)( 7,10)( 8, 9)$
$ 6, 6 $ $20$ $6$ $( 1, 2, 5,12, 3, 4)( 6,11, 8, 7,10, 9)$
$ 10, 2 $ $12$ $10$ $( 1, 2, 5, 6, 9,12, 3, 4, 7, 8)(10,11)$
$ 10, 2 $ $12$ $10$ $( 1, 2, 7, 8,11,12, 3, 6, 9,10)( 4, 5)$
$ 3, 3, 3, 3 $ $20$ $3$ $( 1, 3, 5)( 2, 4,12)( 6,10, 8)( 7,11, 9)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  120.35
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 5A1 5A2 6A 10A1 10A3
Size 1 1 15 15 20 12 12 20 12 12
2 P 1A 1A 1A 1A 3A 5A2 5A1 3A 5A1 5A2
3 P 1A 2A 2B 2C 1A 5A2 5A1 2A 10A3 10A1
5 P 1A 2A 2B 2C 3A 1A 1A 6A 2A 2A
Type
120.35.1a R 1 1 1 1 1 1 1 1 1 1
120.35.1b R 1 −1 −1 1 1 1 1 −1 −1 −1
120.35.3a1 R 3 3 −1 −1 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 −ζ5−2−ζ52 −ζ5−1−ζ5
120.35.3a2 R 3 3 −1 −1 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 −ζ5−1−ζ5 −ζ5−2−ζ52
120.35.3b1 R 3 −3 1 −1 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 ζ5−2+ζ52 ζ5−1+ζ5
120.35.3b2 R 3 −3 1 −1 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 ζ5−1+ζ5 ζ5−2+ζ52
120.35.4a R 4 4 0 0 1 −1 −1 1 −1 −1
120.35.4b R 4 −4 0 0 1 −1 −1 −1 1 1
120.35.5a R 5 5 1 1 −1 0 0 −1 0 0
120.35.5b R 5 −5 −1 1 −1 0 0 1 0 0

magma: CharacterTable(G);