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Properties

Label 24T36
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $Q_8:S_3$

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Show commands: Magma

magma: G := TransitiveGroup(24, 36);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $36$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $Q_8:S_3$
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:  (1,3,18,20,9,11,2,4,17,19,10,12)(5,8,21,24,14,16,6,7,22,23,13,15), (3,23)(4,24)(5,21)(6,22)(7,20)(8,19)(9,17)(10,18)(11,16)(12,15)(13,14)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$8$:  $D_{4}$
$12$:  $D_{6}$
$16$:  $QD_{16}$
$24$:  $(C_6\times C_2):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $D_{6}$

Degree 8: $QD_{16}$

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

Low degree siblings

There are no siblings 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, 2, 1, 1 $ $12$ $2$ $( 3,23)( 4,24)( 5,21)( 6,22)( 7,20)( 8,19)( 9,17)(10,18)(11,16)(12,15)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$
$ 8, 8, 8 $ $6$ $8$ $( 1, 3,13,16, 2, 4,14,15)( 5,24,17,11, 6,23,18,12)( 7, 9,19,21, 8,10,20,22)$
$ 12, 12 $ $4$ $12$ $( 1, 3,18,20, 9,11, 2, 4,17,19,10,12)( 5, 8,21,24,14,16, 6, 7,22,23,13,15)$
$ 8, 8, 8 $ $6$ $8$ $( 1, 4,13,15, 2, 3,14,16)( 5,23,17,12, 6,24,18,11)( 7,10,19,22, 8, 9,20,21)$
$ 12, 12 $ $4$ $12$ $( 1, 5,10,13,17,22, 2, 6, 9,14,18,21)( 3, 7,12,16,19,24, 4, 8,11,15,20,23)$
$ 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1, 7, 2, 8)( 3,21, 4,22)( 5,11, 6,12)( 9,15,10,16)(13,20,14,19)(17,24,18,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 9,17)( 2,10,18)( 3,11,19)( 4,12,20)( 5,14,22)( 6,13,21)( 7,15,24) ( 8,16,23)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$
$ 12, 12 $ $4$ $12$ $( 1,11,10,20,17, 3, 2,12, 9,19,18, 4)( 5,16,13,24,22, 8, 6,15,14,23,21, 7)$
$ 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,13, 2,14)( 3,16, 4,15)( 5,17, 6,18)( 7,19, 8,20)( 9,21,10,22)(11,23,12,24)$

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:  48.17
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A 4A 4B 6A 8A1 8A-1 12A 12B1 12B-1
Size 1 1 12 2 2 4 2 6 6 4 4 4
2 P 1A 1A 1A 3A 2A 2A 3A 4A 4A 6A 6A 6A
3 P 1A 2A 2B 1A 4A 4B 2A 8A1 8A-1 4A 4B 4B
Type
48.17.1a R 1 1 1 1 1 1 1 1 1 1 1 1
48.17.1b R 1 1 −1 1 1 −1 1 1 1 1 −1 −1
48.17.1c R 1 1 −1 1 1 1 1 −1 −1 1 1 1
48.17.1d R 1 1 1 1 1 −1 1 −1 −1 1 −1 −1
48.17.2a R 2 2 0 −1 2 2 −1 0 0 −1 −1 −1
48.17.2b R 2 2 0 2 −2 0 2 0 0 −2 0 0
48.17.2c R 2 2 0 −1 2 −2 −1 0 0 −1 1 1
48.17.2d1 C 2 −2 0 2 0 0 −2 −ζ8−ζ83 ζ8+ζ83 0 0 0
48.17.2d2 C 2 −2 0 2 0 0 −2 ζ8+ζ83 −ζ8−ζ83 0 0 0
48.17.2e1 C 2 2 0 −1 −2 0 −1 0 0 1 −1−2ζ3 1+2ζ3
48.17.2e2 C 2 2 0 −1 −2 0 −1 0 0 1 1+2ζ3 −1−2ζ3
48.17.4a R 4 −4 0 −2 0 0 2 0 0 0 0 0

magma: CharacterTable(G);