↑

Properties

Label 24T21
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $\SL(2,3):C_2$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(24, 21);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $21$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\SL(2,3):C_2$
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:  (1,3,23)(2,4,24)(5,14,22)(6,13,21)(7,10,12)(8,9,11)(15,18,19)(16,17,20), (1,12,10,13,18,15,2,11,9,14,17,16)(3,20,5,21,7,23,4,19,6,22,8,24)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$
$24$:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

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

Degree 8: None

Degree 12: $A_4\times C_2$

Low degree siblings

16T60

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 3, 9)( 4,10)( 5,12)( 6,11)( 7, 8)(15,22)(16,21)(17,24)(18,23)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$
$ 12, 12 $ $4$ $12$ $( 1, 3,11,13,21,17, 2, 4,12,14,22,18)( 5, 7, 9,23,19,16, 6, 8,10,24,20,15)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 3,23)( 2, 4,24)( 5,14,22)( 6,13,21)( 7,10,12)( 8, 9,11)(15,18,19) (16,17,20)$
$ 12, 12 $ $4$ $12$ $( 1, 4,11,14,21,18, 2, 3,12,13,22,17)( 5, 8, 9,24,19,15, 6, 7,10,23,20,16)$
$ 6, 6, 6, 6 $ $4$ $6$ $( 1, 4,23, 2, 3,24)( 5,13,22, 6,14,21)( 7, 9,12, 8,10,11)(15,17,19,16,18,20)$
$ 12, 12 $ $4$ $12$ $( 1, 5, 4,13,23,22, 2, 6, 3,14,24,21)( 7,17, 9,19,12,16, 8,18,10,20,11,15)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 5,16)( 2, 6,15)( 3,19,11)( 4,20,12)( 7,18,22)( 8,17,21)( 9,14,24) (10,13,23)$
$ 12, 12 $ $4$ $12$ $( 1, 6, 4,14,23,21, 2, 5, 3,13,24,22)( 7,18, 9,20,12,15, 8,17,10,19,11,16)$
$ 6, 6, 6, 6 $ $4$ $6$ $( 1, 6,16, 2, 5,15)( 3,20,11, 4,19,12)( 7,17,22, 8,18,21)( 9,13,24,10,14,23)$
$ 4, 4, 4, 4, 4, 4 $ $6$ $4$ $( 1, 7, 2, 8)( 3,22, 4,21)( 5,12, 6,11)( 9,16,10,15)(13,19,14,20)(17,23,18,24)$
$ 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,13, 2,14)( 3,21, 4,22)( 5,23, 6,24)( 7,19, 8,20)( 9,16,10,15)(11,17,12,18)$
$ 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,14, 2,13)( 3,22, 4,21)( 5,24, 6,23)( 7,20, 8,19)( 9,15,10,16)(11,18,12,17)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $48=2^{4} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  48.33
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A1 3A-1 4A1 4A-1 4B 6A1 6A-1 12A1 12A-1 12A5 12A-5
Size 1 1 6 4 4 1 1 6 4 4 4 4 4 4
2 P 1A 1A 1A 3A-1 3A1 2A 2A 2A 3A1 3A-1 6A1 6A-1 6A-1 6A1
3 P 1A 2A 2B 1A 1A 4A-1 4A1 4B 2A 2A 4A1 4A-1 4A1 4A-1
Type
48.33.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
48.33.1b R 1 1 −1 1 1 −1 −1 1 1 1 −1 −1 −1 −1
48.33.1c1 C 1 1 1 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
48.33.1c2 C 1 1 1 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
48.33.1d1 C 1 1 −1 ζ3−1 ζ3 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
48.33.1d2 C 1 1 −1 ζ3 ζ3−1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
48.33.2a1 C 2 −2 0 −1 −1 −2i 2i 0 1 1 −i i −i i
48.33.2a2 C 2 −2 0 −1 −1 2i −2i 0 1 1 i −i i −i
48.33.2b1 C 2 −2 0 −ζ124 ζ122 −2ζ123 2ζ123 0 −ζ122 ζ124 ζ12 −ζ125 ζ125 −ζ12
48.33.2b2 C 2 −2 0 ζ122 −ζ124 2ζ123 −2ζ123 0 ζ124 −ζ122 −ζ125 ζ12 −ζ12 ζ125
48.33.2b3 C 2 −2 0 −ζ124 ζ122 2ζ123 −2ζ123 0 −ζ122 ζ124 −ζ12 ζ125 −ζ125 ζ12
48.33.2b4 C 2 −2 0 ζ122 −ζ124 −2ζ123 2ζ123 0 ζ124 −ζ122 ζ125 −ζ12 ζ12 −ζ125
48.33.3a R 3 3 −1 0 0 3 3 −1 0 0 0 0 0 0
48.33.3b R 3 3 1 0 0 −3 −3 −1 0 0 0 0 0 0

magma: CharacterTable(G);