↑

Properties

Label 24T4538
24T4538 1 12 1->12 23 1->23 2 11 2->11 24 2->24 3 13 3->13 19 3->19 4 14 4->14 20 4->20 5 16 5->16 18 5->18 6 15 6->15 17 6->17 7 9 7->9 21 7->21 8 10 8->10 22 8->22 9->1 9->5 10->2 10->6 11->4 11->5 12->3 12->6 13->2 13->8 14->1 14->7 15->4 15->7 16->3 16->8 17->16 17->23 18->15 18->24 19->12 19->18 20->11 20->17 21->9 21->20 22->10 22->19 23->14 23->21 24->13 24->22
Degree $24$
Order $1536$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_4^2:\GL(2,\mathbb{Z}/4)$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(24, 4538);
 

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 $t$:  $4538$
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:  $(1,12,3,13,2,11,4,14)(5,16,8,10,6,15,7,9)(17,23,21,20)(18,24,22,19)$, $(1,23,14,7,21,9)(2,24,13,8,22,10)(3,19,12,6,17,16)(4,20,11,5,18,15)$
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: $S_4\times C_2$

Degree 8: None

Degree 12: 12T49

Low degree siblings

16T1314 x 2, 24T4479 x 2, 24T4485 x 2, 24T4536 x 2, 24T4538, 24T4558 x 2, 24T4562 x 2, 24T4742 x 2, 24T4747 x 2, 32T96796, 32T96837 x 2, 32T96838 x 2, 32T96839 x 2

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

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

33 x 33 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed