Properties

Label 24T27
24T27 1 20 1->20 23 1->23 2 19 2->19 24 2->24 3 9 3->9 11 3->11 21 3->21 4 10 4->10 12 4->12 22 4->22 5 5->19 5->22 5->24 6 6->20 6->21 6->23 7 13 7->13 17 7->17 8 14 8->14 18 8->18 9->4 16 9->16 9->17 10->3 15 10->15 10->18 11->13 11->17 12->14 12->18 13->8 13->12 14->7 14->11 15->9 15->21 15->24 16->10 16->22 16->23 17->8 17->12 18->7 18->11 19->1 19->6 20->2 20->5 21->4 21->16 22->3 22->15 23->2 23->5 24->1 24->6
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_4\times D_6$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(24, 27);
 
Copy content sage:G = TransitiveGroup(24, 27)
 
Copy content oscar:G = transitive_group(24, 27)
 
Copy content gap:G := TransitiveGroup(24, 27);
 

Group invariants

Abstract group:  $C_4\times D_6$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Copy content oscar:small_group_identification(G)
 
Copy content gap:IdGroup(G);
 
Order:  $48=2^{4} \cdot 3$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Copy content gap:Order(G);
 
Cyclic:  no
Copy content comment:Determine if group is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content oscar:is_cyclic(G)
 
Copy content gap:IsCyclic(G);
 
Abelian:  no
Copy content comment:Determine if group is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content oscar:is_abelian(G)
 
Copy content gap:IsAbelian(G);
 
Solvable:  yes
Copy content comment:Determine if group is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content sage:G.is_solvable()
 
Copy content oscar:is_solvable(G)
 
Copy content gap:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content comment:Nilpotency class
 
Copy content magma:NilpotencyClass(G);
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 
Copy content gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;
 

Group action invariants

Degree $n$:  $24$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Copy content gap:NrMovedPoints(G);
 
Transitive number $t$:  $27$
Copy content comment:Transitive number
 
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Copy content sage:G.transitive_number()
 
Copy content oscar:transitive_group_identification(G)[2]
 
Copy content gap:TransitiveIdentification(G);
 
Parity:  $1$
Copy content comment:Parity
 
Copy content magma:IsEven(G);
 
Copy content sage:all(g.SignPerm() == 1 for g in libgap(G).GeneratorsOfGroup())
 
Copy content oscar:is_even(G)
 
Copy content gap:ForAll(GeneratorsOfGroup(G), g -> SignPerm(g) = 1);
 
Transitivity:  1
Primitive:  no
Copy content comment:Determine if group is primitive
 
Copy content magma:IsPrimitive(G);
 
Copy content sage:G.is_primitive()
 
Copy content oscar:is_primitive(G)
 
Copy content gap:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $8$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(24).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(24), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(24), G));
 
Generators:  $(3,11)(4,12)(5,22)(6,21)(9,17)(10,18)(15,24)(16,23)$, $(1,23,2,24)(3,21,4,22)(5,19,6,20)(7,17,8,18)(9,16,10,15)(11,13,12,14)$, $(1,20,2,19)(3,9,4,10)(5,24,6,23)(7,13,8,14)(11,17,12,18)(15,21,16,22)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 
Copy content gap:GeneratorsOfGroup(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$6$:  $S_3$
$8$:  $C_4\times C_2$ x 6, $C_2^3$
$12$:  $D_{6}$ x 3
$16$:  $C_4\times C_2^2$
$24$:  $S_3 \times C_2^2$, $S_3 \times C_4$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_4$ x 2, $C_2^2$

Degree 6: $D_{6}$ x 3

Degree 8: $C_4\times C_2$

Degree 12: $S_3 \times C_2^2$, $S_3 \times C_4$ x 2

Low degree siblings

24T27 x 3

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 $2^{12}$ $1$ $2$ $12$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5,17)( 6,18)( 7,20)( 8,19)( 9,22)(10,21)(11,23)(12,24)$
2C $2^{12}$ $1$ $2$ $12$ $( 1,13)( 2,14)( 3,15)( 4,16)( 5,18)( 6,17)( 7,19)( 8,20)( 9,21)(10,22)(11,24)(12,23)$
2D $2^{8},1^{8}$ $3$ $2$ $8$ $( 3,11)( 4,12)( 5,22)( 6,21)( 9,17)(10,18)(15,24)(16,23)$
2E $2^{12}$ $3$ $2$ $12$ $( 1,10)( 2, 9)( 3,20)( 4,19)( 5, 6)( 7,16)( 8,15)(11,12)(13,22)(14,21)(17,18)(23,24)$
2F $2^{12}$ $3$ $2$ $12$ $( 1,14)( 2,13)( 3,23)( 4,24)( 5, 9)( 6,10)( 7,20)( 8,19)(11,16)(12,15)(17,22)(18,21)$
2G $2^{12}$ $3$ $2$ $12$ $( 1,21)( 2,22)( 3, 7)( 4, 8)( 5,18)( 6,17)( 9,13)(10,14)(11,24)(12,23)(15,19)(16,20)$
3A $3^{8}$ $2$ $3$ $16$ $( 1, 9,17)( 2,10,18)( 3,11,19)( 4,12,20)( 5,14,22)( 6,13,21)( 7,15,24)( 8,16,23)$
4A1 $4^{6}$ $1$ $4$ $18$ $( 1, 7, 2, 8)( 3,22, 4,21)( 5,12, 6,11)( 9,15,10,16)(13,19,14,20)(17,24,18,23)$
4A-1 $4^{6}$ $1$ $4$ $18$ $( 1, 8, 2, 7)( 3,21, 4,22)( 5,11, 6,12)( 9,16,10,15)(13,20,14,19)(17,23,18,24)$
4B1 $4^{6}$ $1$ $4$ $18$ $( 1,20, 2,19)( 3, 9, 4,10)( 5,24, 6,23)( 7,13, 8,14)(11,17,12,18)(15,21,16,22)$
4B-1 $4^{6}$ $1$ $4$ $18$ $( 1,19, 2,20)( 3,10, 4, 9)( 5,23, 6,24)( 7,14, 8,13)(11,18,12,17)(15,22,16,21)$
4C1 $4^{6}$ $3$ $4$ $18$ $( 1,23, 2,24)( 3,21, 4,22)( 5,19, 6,20)( 7,17, 8,18)( 9,16,10,15)(11,13,12,14)$
4C-1 $4^{6}$ $3$ $4$ $18$ $( 1, 7, 2, 8)( 3, 5, 4, 6)( 9,24,10,23)(11,22,12,21)(13,19,14,20)(15,18,16,17)$
4D1 $4^{6}$ $3$ $4$ $18$ $( 1,11, 2,12)( 3,10, 4, 9)( 5, 8, 6, 7)(13,24,14,23)(15,22,16,21)(17,19,18,20)$
4D-1 $4^{6}$ $3$ $4$ $18$ $( 1,20, 2,19)( 3,17, 4,18)( 5,15, 6,16)( 7,13, 8,14)( 9,12,10,11)(21,23,22,24)$
6A $6^{4}$ $2$ $6$ $20$ $( 1, 5, 9,14,17,22)( 2, 6,10,13,18,21)( 3, 8,11,16,19,23)( 4, 7,12,15,20,24)$
6B $6^{4}$ $2$ $6$ $20$ $( 1,21,17,13, 9, 6)( 2,22,18,14,10, 5)( 3,24,19,15,11, 7)( 4,23,20,16,12, 8)$
6C $6^{4}$ $2$ $6$ $20$ $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$
12A1 $12^{2}$ $2$ $12$ $22$ $( 1,23,10, 7,17,16, 2,24, 9, 8,18,15)( 3,13,12,22,19, 6, 4,14,11,21,20, 5)$
12A-1 $12^{2}$ $2$ $12$ $22$ $( 1,24,10, 8,17,15, 2,23, 9, 7,18,16)( 3,14,12,21,19, 5, 4,13,11,22,20, 6)$
12B1 $12^{2}$ $2$ $12$ $22$ $( 1,11,10,20,17, 3, 2,12, 9,19,18, 4)( 5,16,13,24,22, 8, 6,15,14,23,21, 7)$
12B-1 $12^{2}$ $2$ $12$ $22$ $( 1,12,10,19,17, 4, 2,11, 9,20,18, 3)( 5,15,13,23,22, 7, 6,16,14,24,21, 8)$

Malle's constant $a(G)$:     $1/8$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 
Copy content gap:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 3A 4A1 4A-1 4B1 4B-1 4C1 4C-1 4D1 4D-1 6A 6B 6C 12A1 12A-1 12B1 12B-1
Size 1 1 1 1 3 3 3 3 2 1 1 1 1 3 3 3 3 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A 2A 2A 2A 2A 2A 2A 2A 2A 3A 3A 3A 6C 6C 6C 6C
3 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 4A-1 4A1 4B-1 4B1 4C-1 4C1 4D-1 4D1 2B 2C 2A 4A1 4A-1 4B1 4B-1
Type
48.35.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
48.35.1b R 1 1 −1 −1 −1 −1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 −1 −1 1 1
48.35.1c R 1 1 −1 −1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 1 1 1 −1 −1
48.35.1d R 1 1 −1 −1 1 1 −1 −1 1 −1 −1 1 1 −1 −1 1 1 −1 −1 1 −1 −1 1 1
48.35.1e R 1 1 −1 −1 1 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1 1 −1 −1
48.35.1f R 1 1 1 1 −1 −1 −1 −1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1
48.35.1g R 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1
48.35.1h R 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 −1 −1 −1 −1
48.35.1i1 C 1 −1 −1 1 −1 1 1 −1 1 −i i i −i −i i i −i −1 1 −1 i −i −i i
48.35.1i2 C 1 −1 −1 1 −1 1 1 −1 1 i −i −i i i −i −i i −1 1 −1 −i i i −i
48.35.1j1 C 1 −1 −1 1 1 −1 −1 1 1 −i i i −i i −i −i i −1 1 −1 i −i −i i
48.35.1j2 C 1 −1 −1 1 1 −1 −1 1 1 i −i −i i −i i i −i −1 1 −1 −i i i −i
48.35.1k1 C 1 −1 1 −1 −1 1 −1 1 1 −i i −i i −i i −i i 1 −1 −1 i −i i −i
48.35.1k2 C 1 −1 1 −1 −1 1 −1 1 1 i −i i −i i −i i −i 1 −1 −1 −i i −i i
48.35.1l1 C 1 −1 1 −1 1 −1 1 −1 1 −i i −i i i −i i −i 1 −1 −1 i −i i −i
48.35.1l2 C 1 −1 1 −1 1 −1 1 −1 1 i −i i −i −i i −i i 1 −1 −1 −i i −i i
48.35.2a R 2 2 2 2 0 0 0 0 −1 2 2 2 2 0 0 0 0 −1 −1 −1 −1 −1 −1 −1
48.35.2b R 2 2 −2 −2 0 0 0 0 −1 −2 −2 2 2 0 0 0 0 1 1 −1 1 1 −1 −1
48.35.2c R 2 2 −2 −2 0 0 0 0 −1 2 2 −2 −2 0 0 0 0 1 1 −1 −1 −1 1 1
48.35.2d R 2 2 2 2 0 0 0 0 −1 −2 −2 −2 −2 0 0 0 0 −1 −1 −1 1 1 1 1
48.35.2e1 C 2 −2 −2 2 0 0 0 0 −1 −2i 2i 2i −2i 0 0 0 0 1 −1 1 −i i i −i
48.35.2e2 C 2 −2 −2 2 0 0 0 0 −1 2i −2i −2i 2i 0 0 0 0 1 −1 1 i −i −i i
48.35.2f1 C 2 −2 2 −2 0 0 0 0 −1 −2i 2i −2i 2i 0 0 0 0 −1 1 1 −i i −i i
48.35.2f2 C 2 −2 2 −2 0 0 0 0 −1 2i −2i 2i −2i 0 0 0 0 −1 1 1 i −i i −i

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 
Copy content gap:CharacterTable(G);
 

Regular extensions

Data not computed