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Properties

Label 24T41
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times \SD_{16}$

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Show commands: Magma

magma: G := TransitiveGroup(24, 41);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $41$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times \SD_{16}$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,16,18,7,9,23,2,15,17,8,10,24)(3,6,20,22,11,13,4,5,19,21,12,14), (1,17,9)(2,18,10)(3,8,11,16,19,23)(4,7,12,15,20,24)(5,21,14,6,22,13)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$8$:  $D_{4}$
$12$:  $C_6\times C_2$
$16$:  $QD_{16}$
$24$:  $D_4 \times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: $D_{4}$

Degree 6: $C_6$

Degree 8: $QD_{16}$

Degree 12: $D_4 \times C_3$

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

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:  $3$
Label:  48.26
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A1 3A-1 4A 4B 6A1 6A-1 6B1 6B-1 8A1 8A-1 12A1 12A-1 12B1 12B-1 24A1 24A-1 24A7 24A-7
Size 1 1 4 1 1 2 4 1 1 4 4 2 2 2 2 4 4 2 2 2 2
2 P 1A 1A 1A 3A-1 3A1 2A 2A 3A1 3A-1 3A1 3A-1 4A 4A 6A1 6A-1 6A1 6A-1 12A1 12A-1 12A1 12A-1
3 P 1A 2A 2B 1A 1A 4A 4B 2A 2A 2B 2B 8A1 8A-1 4A 4A 4B 4B 8A-1 8A1 8A1 8A-1
Type
48.26.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
48.26.1b R 1 1 −1 1 1 1 −1 1 1 −1 −1 1 1 1 1 −1 −1 1 1 1 1
48.26.1c R 1 1 −1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1
48.26.1d R 1 1 1 1 1 1 −1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 −1 −1
48.26.1e1 C 1 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1
48.26.1e2 C 1 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
48.26.1f1 C 1 1 −1 ζ3−1 ζ3 1 −1 ζ3 ζ3−1 −ζ3 −ζ3−1 1 1 ζ3−1 ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 ζ3 ζ3−1
48.26.1f2 C 1 1 −1 ζ3 ζ3−1 1 −1 ζ3−1 ζ3 −ζ3−1 −ζ3 1 1 ζ3 ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
48.26.1g1 C 1 1 −1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −1 −1 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1
48.26.1g2 C 1 1 −1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −1 −1 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3
48.26.1h1 C 1 1 1 ζ3−1 ζ3 1 −1 ζ3 ζ3−1 ζ3 ζ3−1 −1 −1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1
48.26.1h2 C 1 1 1 ζ3 ζ3−1 1 −1 ζ3−1 ζ3 ζ3−1 ζ3 −1 −1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3
48.26.2a R 2 2 0 2 2 −2 0 2 2 0 0 0 0 −2 −2 0 0 0 0 0 0
48.26.2b1 C 2 −2 0 2 2 0 0 −2 −2 0 0 −ζ8−ζ83 ζ8+ζ83 0 0 0 0 −ζ8−ζ83 ζ8+ζ83 ζ8+ζ83 −ζ8−ζ83
48.26.2b2 C 2 −2 0 2 2 0 0 −2 −2 0 0 ζ8+ζ83 −ζ8−ζ83 0 0 0 0 ζ8+ζ83 −ζ8−ζ83 −ζ8−ζ83 ζ8+ζ83
48.26.2c1 C 2 2 0 2ζ3−1 2ζ3 −2 0 2ζ3 2ζ3−1 0 0 0 0 −2ζ3−1 −2ζ3 0 0 0 0 0 0
48.26.2c2 C 2 2 0 2ζ3 2ζ3−1 −2 0 2ζ3−1 2ζ3 0 0 0 0 −2ζ3 −2ζ3−1 0 0 0 0 0 0
48.26.2d1 C 2 −2 0 −2ζ244 2ζ248 0 0 −2ζ248 2ζ244 0 0 ζ24−ζ243−ζ245 −ζ24+ζ243+ζ245 0 0 0 0 ζ243+ζ245−ζ247 ζ24−ζ247 −ζ243−ζ245+ζ247 −ζ24+ζ247
48.26.2d2 C 2 −2 0 2ζ248 −2ζ244 0 0 2ζ244 −2ζ248 0 0 −ζ24+ζ243+ζ245 ζ24−ζ243−ζ245 0 0 0 0 ζ24−ζ247 ζ243+ζ245−ζ247 −ζ24+ζ247 −ζ243−ζ245+ζ247
48.26.2d3 C 2 −2 0 −2ζ244 2ζ248 0 0 −2ζ248 2ζ244 0 0 −ζ24+ζ243+ζ245 ζ24−ζ243−ζ245 0 0 0 0 −ζ243−ζ245+ζ247 −ζ24+ζ247 ζ243+ζ245−ζ247 ζ24−ζ247
48.26.2d4 C 2 −2 0 2ζ248 −2ζ244 0 0 2ζ244 −2ζ248 0 0 ζ24−ζ243−ζ245 −ζ24+ζ243+ζ245 0 0 0 0 −ζ24+ζ247 −ζ243−ζ245+ζ247 ζ24−ζ247 ζ243+ζ245−ζ247

magma: CharacterTable(G);