Properties

Label 24T4563
24T4563 1 7 1->7 13 1->13 2 8 2->8 14 2->14 3 5 3->5 16 3->16 4 6 4->6 15 4->15 5->4 10 5->10 6->3 9 6->9 12 7->12 11 8->11 19 9->19 21 9->21 20 10->20 22 10->22 18 11->18 24 11->24 17 12->17 23 12->23 13->18 13->23 14->17 14->24 15->20 15->22 16->19 16->21 17->5 17->12 18->6 18->11 19->8 19->10 20->7 20->9 21->1 21->16 22->2 22->15 23->3 23->13 24->4 24->14
Degree $24$
Order $1536$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_4^2.\GL(2,\mathbb{Z}/4)$

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Copy content magma:G := TransitiveGroup(24, 4563);
 

Group invariants

Abstract group:  $C_4^2.\GL(2,\mathbb{Z}/4)$
Copy content magma:IdentifyGroup(G);
 
Order:  $1536=2^{9} \cdot 3$
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 $t$:  $4563$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,7)(2,8)(3,5,4,6)(9,21,16,19,10,22,15,20)(11,24,14,17,12,23,13,18)$, $(1,13,23,3,16,21)(2,14,24,4,15,22)(5,10,20,7,12,17)(6,9,19,8,11,18)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$8$:  $D_{4}$
$12$:  $D_{6}$
$24$:  $S_4$ x 3, $(C_6\times C_2):C_2$
$48$:  $S_4\times C_2$ x 3
$96$:  $V_4^2:S_3$, 12T49 x 3
$192$:  12T100
$384$:  12T135

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4$, $S_4\times C_2$ x 2

Degree 8: None

Degree 12: 12T101

Low degree siblings

24T4534 x 2, 24T4537 x 2, 24T4554 x 2, 24T4563, 24T4740 x 4, 24T4745 x 4

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 $2^{8},1^{8}$ $3$ $2$ $8$ $( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$
2B $2^{6},1^{12}$ $4$ $2$ $6$ $( 1, 2)( 3, 4)(13,14)(15,16)(17,18)(19,20)$
2C $2^{12}$ $8$ $2$ $12$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,24)(22,23)$
2D $2^{8},1^{8}$ $12$ $2$ $8$ $( 9,15)(10,16)(11,14)(12,13)(17,21)(18,22)(19,24)(20,23)$
2E $2^{12}$ $12$ $2$ $12$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,16)(10,15)(11,13)(12,14)(17,21)(18,22)(19,24)(20,23)$
2F $2^{10},1^{4}$ $12$ $2$ $10$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,15)(10,16)(11,14)(12,13)(19,20)(23,24)$
2G $2^{10},1^{4}$ $12$ $2$ $10$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,10)(15,16)(17,22)(18,21)(19,23)(20,24)$
2H $2^{8},1^{8}$ $24$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)(11,12)(15,16)(19,20)(21,22)$
2I $2^{12}$ $24$ $2$ $12$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,13)(10,14)(11,16)(12,15)(17,24)(18,23)(19,21)(20,22)$
2J $2^{11},1^{2}$ $48$ $2$ $11$ $( 1,16)( 2,15)( 3,13)( 4,14)( 5,11)( 6,12)( 7,10)( 8, 9)(17,18)(21,22)(23,24)$
3A $3^{8}$ $128$ $3$ $16$ $( 1,21,12)( 2,22,11)( 3,23,10)( 4,24, 9)( 5,17,16)( 6,18,15)( 7,20,13)( 8,19,14)$
4A $4^{4},2^{4}$ $6$ $4$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12,10,11)(13,15,14,16)(17,20,18,19)(21,24,22,23)$
4B $4^{4},1^{8}$ $6$ $4$ $12$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)$
4C $4^{4},2^{2},1^{4}$ $12$ $4$ $14$ $( 1, 4, 2, 3)( 5, 8, 6, 7)(13,14)(15,16)(17,19,18,20)(21,23,22,24)$
4D $4^{4},2^{2},1^{4}$ $12$ $4$ $14$ $( 1, 8, 2, 7)( 3, 5, 4, 6)( 9,15,10,16)(11,14,12,13)(17,18)(23,24)$
4E $4^{4},2^{2},1^{4}$ $12$ $4$ $14$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,14,10,13)(11,16,12,15)(17,18)(21,22)$
4F $4^{4},2^{2},1^{4}$ $24$ $4$ $14$ $( 1, 2)( 3, 4)( 9,15,10,16)(11,14,12,13)(17,22,18,21)(19,23,20,24)$
4G $4^{4},2^{4}$ $24$ $4$ $16$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,16,10,15)(11,14,12,13)(17,22,18,21)(19,24,20,23)$
4H $4^{4},2^{4}$ $24$ $4$ $16$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,14)(10,13)(11,16)(12,15)(17,22,18,21)(19,23,20,24)$
4I $4^{4},2^{4}$ $24$ $4$ $16$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,15)(10,16)(11,14)(12,13)(17,23,18,24)(19,21,20,22)$
4J $4^{4},2,1^{6}$ $48$ $4$ $13$ $( 1,14, 2,13)( 3,16, 4,15)( 5, 9, 6,10)( 7,11, 8,12)(19,20)$
4K $4^{3},2^{6}$ $96$ $4$ $15$ $( 1,17, 2,18)( 3,20, 4,19)( 5,21)( 6,22)( 7,24)( 8,23)( 9,11)(10,12)(13,16,14,15)$
4L $4^{5},2^{2}$ $96$ $4$ $17$ $( 1,13, 7,12)( 2,14, 8,11)( 3,15, 6, 9)( 4,16, 5,10)(17,22)(18,21)(19,23,20,24)$
4M $4^{5},2^{2}$ $96$ $4$ $17$ $( 1,21, 6,18)( 2,22, 5,17)( 3,24, 7,20)( 4,23, 8,19)( 9,15,10,16)(11,14)(12,13)$
6A $6^{2},3^{4}$ $128$ $6$ $18$ $( 1,11,21, 2,12,22)( 3, 9,23, 4,10,24)( 5,16,17)( 6,15,18)( 7,13,20)( 8,14,19)$
6B1 $6^{4}$ $128$ $6$ $20$ $( 1,18,11, 4,20, 9)( 2,17,12, 3,19,10)( 5,21,16, 8,24,13)( 6,22,15, 7,23,14)$
6B-1 $6^{4}$ $128$ $6$ $20$ $( 1, 9,20, 4,11,18)( 2,10,19, 3,12,17)( 5,13,24, 8,16,21)( 6,14,23, 7,15,22)$
8A $8^{2},2,1^{6}$ $48$ $8$ $15$ $( 1,15, 4,14, 2,16, 3,13)( 5,11, 8,10, 6,12, 7, 9)(23,24)$
8B $8^{2},2^{3},1^{2}$ $48$ $8$ $17$ $( 1,14, 3,15, 2,13, 4,16)( 5, 9, 7,12, 6,10, 8,11)(17,18)(19,20)(23,24)$
8C $8^{2},4,2^{2}$ $96$ $8$ $19$ $( 1,23, 4,22, 2,24, 3,21)( 5,19, 8,18, 6,20, 7,17)( 9,12)(10,11)(13,16,14,15)$
8D $8^{2},4,2^{2}$ $96$ $8$ $19$ $( 1, 9, 8,15, 2,10, 7,16)( 3,12, 5,13, 4,11, 6,14)(17,23,18,24)(19,21)(20,22)$
8E $8^{2},4,2^{2}$ $96$ $8$ $19$ $( 1,11, 6,16, 2,12, 5,15)( 3, 9, 8,14, 4,10, 7,13)(17,21,18,22)(19,24)(20,23)$

Malle's constant $a(G)$:     $1/6$

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Character table

33 x 33 character table

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Regular extensions

Data not computed