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Properties

Label 24T89
24T89 1 12 1->12 17 1->17 2 11 2->11 18 2->18 3 14 3->14 20 3->20 4 13 4->13 19 4->19 5 15 5->15 22 5->22 6 16 6->16 21 6->21 7 10 7->10 23 7->23 8 9 8->9 24 8->24 9->4 9->6 10->3 10->5 11->6 11->7 12->5 12->8 13->1 13->7 14->2 14->8 15->2 15->4 16->1 16->3 17->14 17->24 18->13 18->23 19->16 19->17 20->15 20->18 21->10 21->20 22->9 22->19 23->12 23->22 24->11 24->21
Degree $24$
Order $96$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $A_4:C_8$

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Copy content magma:G := TransitiveGroup(24, 89);
 

Group invariants

Abstract group:  $A_4:C_8$
Copy content magma:IdentifyGroup(G);
 
Order:  $96=2^{5} \cdot 3$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $24$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $89$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $8$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,12,5,15,2,11,6,16)(3,14,8,9,4,13,7,10)(17,24,21,20,18,23,22,19)$, $(1,17,14,2,18,13)(3,20,15,4,19,16)(5,22,9,6,21,10)(7,23,12,8,24,11)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$6$:  $S_3$
$8$:  $C_8$
$12$:  $C_3 : C_4$
$24$:  $S_4$, 24T8
$48$:  12T27

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$

Degree 6: $S_3$

Degree 8: None

Degree 12: $C_3 : C_4$

Low degree siblings

32T413

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{24}$ $1$ $1$ $0$ $()$
2A $2^{12}$ $1$ $2$ $12$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$
2B $2^{8},1^{8}$ $3$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2C $2^{4},1^{16}$ $3$ $2$ $4$ $(17,18)(19,20)(21,22)(23,24)$
3A $3^{8}$ $8$ $3$ $16$ $( 1,17,13)( 2,18,14)( 3,20,16)( 4,19,15)( 5,22,10)( 6,21, 9)( 7,23,11)( 8,24,12)$
4A1 $4^{6}$ $1$ $4$ $18$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,14,10,13)(11,15,12,16)(17,21,18,22)(19,24,20,23)$
4A-1 $4^{6}$ $1$ $4$ $18$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,13,10,14)(11,16,12,15)(17,22,18,21)(19,23,20,24)$
4B1 $4^{6}$ $3$ $4$ $18$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,13,10,14)(11,16,12,15)(17,21,18,22)(19,24,20,23)$
4B-1 $4^{6}$ $3$ $4$ $18$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,14,10,13)(11,15,12,16)(17,22,18,21)(19,23,20,24)$
6A $6^{4}$ $8$ $6$ $20$ $( 1,14,17, 2,13,18)( 3,15,20, 4,16,19)( 5, 9,22, 6,10,21)( 7,12,23, 8,11,24)$
8A1 $8^{3}$ $6$ $8$ $21$ $( 1,11, 6,15, 2,12, 5,16)( 3,13, 7, 9, 4,14, 8,10)(17,23,21,19,18,24,22,20)$
8A-1 $8^{3}$ $6$ $8$ $21$ $( 1,16, 5,12, 2,15, 6,11)( 3,10, 8,14, 4, 9, 7,13)(17,20,22,24,18,19,21,23)$
8A3 $8^{3}$ $6$ $8$ $21$ $( 1,15, 5,11, 2,16, 6,12)( 3, 9, 8,13, 4,10, 7,14)(17,19,22,23,18,20,21,24)$
8A-3 $8^{3}$ $6$ $8$ $21$ $( 1,12, 6,16, 2,11, 5,15)( 3,14, 7,10, 4,13, 8, 9)(17,24,21,20,18,23,22,19)$
8B1 $8^{3}$ $6$ $8$ $21$ $( 1,12, 5,15, 2,11, 6,16)( 3,14, 8, 9, 4,13, 7,10)(17,24,21,20,18,23,22,19)$
8B-1 $8^{3}$ $6$ $8$ $21$ $( 1,15, 6,12, 2,16, 5,11)( 3, 9, 7,14, 4,10, 8,13)(17,19,22,23,18,20,21,24)$
8B3 $8^{3}$ $6$ $8$ $21$ $( 1,16, 6,11, 2,15, 5,12)( 3,10, 7,13, 4, 9, 8,14)(17,20,22,24,18,19,21,23)$
8B-3 $8^{3}$ $6$ $8$ $21$ $( 1,11, 5,16, 2,12, 6,15)( 3,13, 8,10, 4,14, 7, 9)(17,23,21,19,18,24,22,20)$
12A1 $12^{2}$ $8$ $12$ $22$ $( 1,21,14, 5,17, 9, 2,22,13, 6,18,10)( 3,23,15, 8,20,11, 4,24,16, 7,19,12)$
12A-1 $12^{2}$ $8$ $12$ $22$ $( 1,22,14, 6,17,10, 2,21,13, 5,18, 9)( 3,24,15, 7,20,12, 4,23,16, 8,19,11)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 6A 8A1 8A-1 8A3 8A-3 8B1 8B-1 8B3 8B-3 12A1 12A-1
Size 1 1 3 3 8 1 1 3 3 8 6 6 6 6 6 6 6 6 8 8
2 P 1A 1A 1A 1A 3A 2A 2A 2A 2A 3A 4A1 4A-1 4A-1 4A1 4B1 4B-1 4B-1 4B1 6A 6A
3 P 1A 2A 2B 2C 1A 4A-1 4A1 4B-1 4B1 2A 8A3 8A-3 8A1 8A-1 8B3 8B-3 8B1 8B-1 4A-1 4A1
Type
96.65.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
96.65.1b R 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1
96.65.1c1 C 1 1 1 1 1 −1 −1 −1 −1 1 −i −i i i i −i i −i −1 −1
96.65.1c2 C 1 1 1 1 1 −1 −1 −1 −1 1 i i −i −i −i i −i i −1 −1
96.65.1d1 C 1 −1 1 −1 1 −ζ82 ζ82 −ζ82 ζ82 −1 −ζ8 −ζ8 ζ83 −ζ83 ζ83 ζ8 −ζ83 ζ8 −ζ82 ζ82
96.65.1d2 C 1 −1 1 −1 1 ζ82 −ζ82 ζ82 −ζ82 −1 ζ83 ζ83 −ζ8 ζ8 −ζ8 −ζ83 ζ8 −ζ83 ζ82 −ζ82
96.65.1d3 C 1 −1 1 −1 1 −ζ82 ζ82 −ζ82 ζ82 −1 ζ8 ζ8 −ζ83 ζ83 −ζ83 −ζ8 ζ83 −ζ8 −ζ82 ζ82
96.65.1d4 C 1 −1 1 −1 1 ζ82 −ζ82 ζ82 −ζ82 −1 −ζ83 −ζ83 ζ8 −ζ8 ζ8 ζ83 −ζ8 ζ83 ζ82 −ζ82
96.65.2a R 2 2 2 2 −1 2 2 2 2 −1 0 0 0 0 0 0 0 0 −1 −1
96.65.2b S 2 2 2 2 −1 −2 −2 −2 −2 −1 0 0 0 0 0 0 0 0 1 1
96.65.2c1 C 2 −2 2 −2 −1 −2i 2i −2i 2i 1 0 0 0 0 0 0 0 0 i −i
96.65.2c2 C 2 −2 2 −2 −1 2i −2i 2i −2i 1 0 0 0 0 0 0 0 0 −i i
96.65.3a R 3 3 −1 −1 0 3 3 −1 −1 0 −1 1 −1 1 1 1 −1 −1 0 0
96.65.3b R 3 3 −1 −1 0 3 3 −1 −1 0 1 −1 1 −1 −1 −1 1 1 0 0
96.65.3c1 C 3 3 −1 −1 0 −3 −3 1 1 0 −i i i −i −i i i −i 0 0
96.65.3c2 C 3 3 −1 −1 0 −3 −3 1 1 0 i −i −i i i −i −i i 0 0
96.65.3d1 C 3 −3 −1 1 0 −3ζ82 3ζ82 ζ82 −ζ82 0 −ζ8 ζ8 ζ83 ζ83 −ζ83 −ζ8 −ζ83 ζ8 0 0
96.65.3d2 C 3 −3 −1 1 0 3ζ82 −3ζ82 −ζ82 ζ82 0 ζ83 −ζ83 −ζ8 −ζ8 ζ8 ζ83 ζ8 −ζ83 0 0
96.65.3d3 C 3 −3 −1 1 0 −3ζ82 3ζ82 ζ82 −ζ82 0 ζ8 −ζ8 −ζ83 −ζ83 ζ83 ζ8 ζ83 −ζ8 0 0
96.65.3d4 C 3 −3 −1 1 0 3ζ82 −3ζ82 −ζ82 ζ82 0 −ζ83 ζ83 ζ8 ζ8 −ζ8 −ζ83 −ζ8 ζ83 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed