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Properties

Label 32T9389
32T9389 1 4 1->4 13 1->13 2 3 2->3 14 2->14 3->1 15 3->15 4->2 16 4->16 5 8 5->8 20 5->20 6 7 6->7 19 6->19 7->5 18 7->18 8->6 17 8->17 9 12 9->12 27 9->27 10 11 10->11 28 10->28 11->9 26 11->26 12->10 25 12->25 13->26 29 13->29 14->25 30 14->30 15->27 32 15->32 16->28 31 16->31 17->6 22 17->22 18->5 21 18->21 19->7 23 19->23 20->8 24 20->24 21->10 21->17 22->9 22->18 23->11 23->20 24->12 24->19 25->1 25->13 26->2 26->14 27->4 27->16 28->3 28->15 29->21 29->32 30->22 30->31 31->24 31->29 32->23 32->30
Degree $32$
Order $384$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $(C_2^2\times C_8):A_4$

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Copy content magma:G := TransitiveGroup(32, 9389);
 

Group invariants

Abstract group:  $(C_2^2\times C_8):A_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $384=2^{7} \cdot 3$
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 $n$:  $32$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $9389$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,13,29,21,10,28,3,15,32,23,11,26,2,14,30,22,9,27,4,16,31,24,12,25)(5,20,8,17,6,19,7,18)$, $(1,4,2,3)(5,8,6,7)(9,12,10,11)(13,26,14,25)(15,27,16,28)(17,22,18,21)(19,23,20,24)(29,32,30,31)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$12$:  $A_4$, $C_{12}$
$24$:  $A_4\times C_2$
$48$:  12T29
$96$:  $C_2^4:C_6$
$192$:  12T99

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $C_2^4:C_6$

Degree 16: 16T418

Low degree siblings

32T9530

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{32}$ $1$ $1$ $0$ $()$
2A $2^{16}$ $1$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2B $2^{16}$ $6$ $2$ $16$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,32)(10,31)(11,29)(12,30)(13,27)(14,28)(15,25)(16,26)(17,23)(18,24)(19,21)(20,22)$
2C $2^{12},1^{8}$ $12$ $2$ $12$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,29)( 6,30)( 7,32)( 8,31)(13,14)(15,16)(21,22)(23,24)$
2D $2^{16}$ $12$ $2$ $16$ $( 1,29)( 2,30)( 3,32)( 4,31)( 5, 9)( 6,10)( 7,12)( 8,11)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28)$
3A1 $3^{8},1^{8}$ $16$ $3$ $16$ $( 5,11,29)( 6,12,30)( 7, 9,32)( 8,10,31)(13,24,27)(14,23,28)(15,21,25)(16,22,26)$
3A-1 $3^{8},1^{8}$ $16$ $3$ $16$ $( 5,29,11)( 6,30,12)( 7,32, 9)( 8,31,10)(13,27,24)(14,28,23)(15,25,21)(16,26,22)$
4A1 $4^{8}$ $1$ $4$ $24$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)(17,20,18,19)(21,23,22,24)(25,28,26,27)(29,32,30,31)$
4A-1 $4^{8}$ $1$ $4$ $24$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)(17,19,18,20)(21,24,22,23)(25,27,26,28)(29,31,30,32)$
4B $4^{8}$ $6$ $4$ $24$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,30,10,29)(11,32,12,31)(13,25,14,26)(15,28,16,27)(17,22,18,21)(19,23,20,24)$
4C $4^{8}$ $12$ $4$ $24$ $( 1,31, 2,32)( 3,29, 4,30)( 5,11, 6,12)( 7, 9, 8,10)(13,24,14,23)(15,21,16,22)(17,27,18,28)(19,26,20,25)$
4D $4^{8}$ $12$ $4$ $24$ $( 1,12, 2,11)( 3,10, 4, 9)( 5,32, 6,31)( 7,30, 8,29)(13,16,14,15)(17,20,18,19)(21,24,22,23)(25,28,26,27)$
6A1 $6^{4},2^{4}$ $16$ $6$ $24$ $( 1, 2)( 3, 4)( 5,30,11, 6,29,12)( 7,31, 9, 8,32,10)(13,28,24,14,27,23)(15,26,21,16,25,22)(17,18)(19,20)$
6A-1 $6^{4},2^{4}$ $16$ $6$ $24$ $( 1, 2)( 3, 4)( 5,12,29, 6,11,30)( 7,10,32, 8, 9,31)(13,23,27,14,24,28)(15,22,25,16,21,26)(17,18)(19,20)$
8A1 $8^{4}$ $4$ $8$ $28$ $( 1,19, 3,17, 2,20, 4,18)( 5,23, 7,22, 6,24, 8,21)( 9,26,12,27,10,25,11,28)(13,31,15,29,14,32,16,30)$
8A-1 $8^{4}$ $4$ $8$ $28$ $( 1,18, 4,20, 2,17, 3,19)( 5,21, 8,24, 6,22, 7,23)( 9,28,11,25,10,27,12,26)(13,30,16,32,14,29,15,31)$
8A3 $8^{4}$ $4$ $8$ $28$ $( 1,17, 4,19, 2,18, 3,20)( 5,22, 8,23, 6,21, 7,24)( 9,27,11,26,10,28,12,25)(13,29,16,31,14,30,15,32)$
8A-3 $8^{4}$ $4$ $8$ $28$ $( 1,20, 3,18, 2,19, 4,17)( 5,24, 7,21, 6,23, 8,22)( 9,25,12,28,10,26,11,27)(13,32,15,30,14,31,16,29)$
8B1 $8^{4}$ $24$ $8$ $28$ $( 1,16, 5,27, 2,15, 6,28)( 3,13, 7,25, 4,14, 8,26)( 9,22,30,18,10,21,29,17)(11,23,32,20,12,24,31,19)$
8B-1 $8^{4}$ $24$ $8$ $28$ $( 1,28, 6,15, 2,27, 5,16)( 3,26, 8,14, 4,25, 7,13)( 9,17,29,21,10,18,30,22)(11,19,31,24,12,20,32,23)$
12A1 $12^{2},4^{2}$ $16$ $12$ $28$ $( 1, 4, 2, 3)( 5,10,30, 7,11,31, 6, 9,29, 8,12,32)(13,22,28,15,24,26,14,21,27,16,23,25)(17,19,18,20)$
12A-1 $12^{2},4^{2}$ $16$ $12$ $28$ $( 1, 3, 2, 4)( 5,32,12, 8,29, 9, 6,31,11, 7,30,10)(13,25,23,16,27,21,14,26,24,15,28,22)(17,20,18,19)$
12A5 $12^{2},4^{2}$ $16$ $12$ $28$ $( 1, 4, 2, 3)( 5,31,12, 7,29,10, 6,32,11, 8,30, 9)(13,26,23,15,27,22,14,25,24,16,28,21)(17,19,18,20)$
12A-5 $12^{2},4^{2}$ $16$ $12$ $28$ $( 1, 3, 2, 4)( 5, 9,30, 8,11,32, 6,10,29, 7,12,31)(13,21,28,16,24,25,14,22,27,15,23,26)(17,20,18,19)$
24A1 $24,8$ $16$ $24$ $30$ $( 1,17, 4,19, 2,18, 3,20)( 5,16,10,23,30,25, 7,13,11,22,31,28, 6,15, 9,24,29,26, 8,14,12,21,32,27)$
24A-1 $24,8$ $16$ $24$ $30$ $( 1,20, 3,18, 2,19, 4,17)( 5,27,32,21,12,14, 8,26,29,24, 9,15, 6,28,31,22,11,13, 7,25,30,23,10,16)$
24A5 $24,8$ $16$ $24$ $30$ $( 1,18, 4,20, 2,17, 3,19)( 5,25,31,24,12,16, 7,28,29,21,10,13, 6,26,32,23,11,15, 8,27,30,22, 9,14)$
24A-5 $24,8$ $16$ $24$ $30$ $( 1,19, 3,17, 2,20, 4,18)( 5,14, 9,22,30,27, 8,15,11,23,32,26, 6,13,10,21,29,28, 7,16,12,24,31,25)$
24A7 $24,8$ $16$ $24$ $30$ $( 1,20, 3,18, 2,19, 4,17)( 5,13, 9,21,30,28, 8,16,11,24,32,25, 6,14,10,22,29,27, 7,15,12,23,31,26)$
24A-7 $24,8$ $16$ $24$ $30$ $( 1,17, 4,19, 2,18, 3,20)( 5,26,31,23,12,15, 7,27,29,22,10,14, 6,25,32,24,11,16, 8,28,30,21, 9,13)$
24A11 $24,8$ $16$ $24$ $30$ $( 1,19, 3,17, 2,20, 4,18)( 5,28,32,22,12,13, 8,25,29,23, 9,16, 6,27,31,21,11,14, 7,26,30,24,10,15)$
24A-11 $24,8$ $16$ $24$ $30$ $( 1,18, 4,20, 2,17, 3,19)( 5,15,10,24,30,26, 7,14,11,21,31,27, 6,16, 9,23,29,25, 8,13,12,22,32,28)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

32 x 32 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed