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Properties

Label 32T9549
32T9549 1 5 1->5 28 1->28 2 6 2->6 27 2->27 3 18 3->18 23 3->23 4 17 4->17 24 4->24 9 5->9 13 5->13 10 6->10 14 6->14 7 7->17 21 7->21 8 8->18 22 8->22 20 9->20 29 9->29 19 10->19 30 10->30 11 11->21 26 11->26 12 12->22 25 12->25 13->1 14->2 15 15->25 31 15->31 16 16->26 32 16->32 17->31 18->32 19->15 20->16 23->19 23->29 24->20 24->30 25->13 26->14 27->7 28->8 29->27 30->28 31->3 32->4
Degree $32$
Order $448$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_4\times F_8$

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Copy content magma:G := TransitiveGroup(32, 9549);
 

Group invariants

Abstract group:  $D_4\times F_8$
Copy content magma:IdentifyGroup(G);
 
Order:  $448=2^{6} \cdot 7$
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 $t$:  $9549$
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:  $(1,5,9,29,27,7,17,31,3,23,19,15,25,13)(2,6,10,30,28,8,18,32,4,24,20,16,26,14)(11,21)(12,22)$, $(1,28)(2,27)(3,18)(4,17)(5,13)(6,14)(7,21)(8,22)(9,20)(10,19)(11,26)(12,25)(15,31)(16,32)(23,29)(24,30)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$7$:  $C_7$
$8$:  $D_{4}$
$14$:  $C_{14}$ x 3
$28$:  28T2
$56$:  $C_2^3:C_7$, 28T5
$112$:  14T9 x 3
$224$:  28T38

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $C_2^3:C_7$

Degree 16: 16T196

Low degree siblings

28T59 x 2, 32T9549

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 $2^{8},1^{16}$ $2$ $2$ $8$ $( 5, 6)( 7, 8)(13,14)(15,16)(21,22)(23,24)(29,30)(31,32)$
2C $2^{16}$ $2$ $2$ $16$ $( 1,32)( 2,31)( 3, 6)( 4, 5)( 7,26)( 8,25)( 9,24)(10,23)(11,22)(12,21)(13,18)(14,17)(15,28)(16,27)(19,30)(20,29)$
2D $2^{16}$ $7$ $2$ $16$ $( 1,25)( 2,26)( 3, 9)( 4,10)( 5,23)( 6,24)( 7,31)( 8,32)(11,27)(12,28)(13,29)(14,30)(15,21)(16,22)(17,19)(18,20)$
2E $2^{16}$ $7$ $2$ $16$ $( 1,26)( 2,25)( 3,10)( 4, 9)( 5,24)( 6,23)( 7,32)( 8,31)(11,28)(12,27)(13,30)(14,29)(15,22)(16,21)(17,20)(18,19)$
2F $2^{16}$ $14$ $2$ $16$ $( 1,25)( 2,26)( 3, 9)( 4,10)( 5,24)( 6,23)( 7,32)( 8,31)(11,27)(12,28)(13,30)(14,29)(15,22)(16,21)(17,19)(18,20)$
2G $2^{16}$ $14$ $2$ $16$ $( 1, 8)( 2, 7)( 3,24)( 4,23)( 5,10)( 6, 9)(11,16)(12,15)(13,20)(14,19)(17,30)(18,29)(21,28)(22,27)(25,32)(26,31)$
4A $4^{8}$ $2$ $4$ $24$ $( 1,31, 2,32)( 3, 5, 4, 6)( 7,26, 8,25)( 9,23,10,24)(11,21,12,22)(13,18,14,17)(15,28,16,27)(19,29,20,30)$
4B $4^{8}$ $14$ $4$ $24$ $( 1, 8, 2, 7)( 3,24, 4,23)( 5, 9, 6,10)(11,16,12,15)(13,19,14,20)(17,30,18,29)(21,27,22,28)(25,32,26,31)$
7A1 $7^{4},1^{4}$ $8$ $7$ $24$ $( 3, 9,17,27,19,25,11)( 4,10,18,28,20,26,12)( 5,23,13,15,29, 7,21)( 6,24,14,16,30, 8,22)$
7A-1 $7^{4},1^{4}$ $8$ $7$ $24$ $( 3,11,25,19,27,17, 9)( 4,12,26,20,28,18,10)( 5,21, 7,29,15,13,23)( 6,22, 8,30,16,14,24)$
7A2 $7^{4},1^{4}$ $8$ $7$ $24$ $( 3,17,19,11, 9,27,25)( 4,18,20,12,10,28,26)( 5,13,29,21,23,15, 7)( 6,14,30,22,24,16, 8)$
7A-2 $7^{4},1^{4}$ $8$ $7$ $24$ $( 3,25,27, 9,11,19,17)( 4,26,28,10,12,20,18)( 5, 7,15,23,21,29,13)( 6, 8,16,24,22,30,14)$
7A3 $7^{4},1^{4}$ $8$ $7$ $24$ $( 3,27,11,17,25, 9,19)( 4,28,12,18,26,10,20)( 5,15,21,13, 7,23,29)( 6,16,22,14, 8,24,30)$
7A-3 $7^{4},1^{4}$ $8$ $7$ $24$ $( 3,19, 9,25,17,11,27)( 4,20,10,26,18,12,28)( 5,29,23, 7,13,21,15)( 6,30,24, 8,14,22,16)$
14A1 $14^{2},2^{2}$ $8$ $14$ $28$ $( 1, 2)( 3,10,17,28,19,26,11, 4, 9,18,27,20,25,12)( 5,24,13,16,29, 8,21, 6,23,14,15,30, 7,22)(31,32)$
14A-1 $14^{2},2^{2}$ $8$ $14$ $28$ $( 1, 2)( 3,12,25,20,27,18, 9, 4,11,26,19,28,17,10)( 5,22, 7,30,15,14,23, 6,21, 8,29,16,13,24)(31,32)$
14A3 $14^{2},2^{2}$ $8$ $14$ $28$ $( 1, 2)( 3,28,11,18,25,10,19, 4,27,12,17,26, 9,20)( 5,16,21,14, 7,24,29, 6,15,22,13, 8,23,30)(31,32)$
14A-3 $14^{2},2^{2}$ $8$ $14$ $28$ $( 1, 2)( 3,20, 9,26,17,12,27, 4,19,10,25,18,11,28)( 5,30,23, 8,13,22,15, 6,29,24, 7,14,21,16)(31,32)$
14A5 $14^{2},2^{2}$ $8$ $14$ $28$ $( 1, 2)( 3,26,27,10,11,20,17, 4,25,28, 9,12,19,18)( 5, 8,15,24,21,30,13, 6, 7,16,23,22,29,14)(31,32)$
14A-5 $14^{2},2^{2}$ $8$ $14$ $28$ $( 1, 2)( 3,18,19,12, 9,28,25, 4,17,20,11,10,27,26)( 5,14,29,22,23,16, 7, 6,13,30,21,24,15, 8)(31,32)$
14B1 $14,7^{2},2,1^{2}$ $16$ $14$ $26$ $( 3,19, 9,25,17,11,27)( 4,20,10,26,18,12,28)( 5,30,23, 8,13,22,15, 6,29,24, 7,14,21,16)(31,32)$
14B-1 $14,7^{2},2,1^{2}$ $16$ $14$ $26$ $( 3,27,11,17,25, 9,19)( 4,28,12,18,26,10,20)( 5,16,21,14, 7,24,29, 6,15,22,13, 8,23,30)(31,32)$
14B3 $14,7^{2},2,1^{2}$ $16$ $14$ $26$ $( 3,25,27, 9,11,19,17)( 4,26,28,10,12,20,18)( 5, 8,15,24,21,30,13, 6, 7,16,23,22,29,14)(31,32)$
14B-3 $14,7^{2},2,1^{2}$ $16$ $14$ $26$ $( 3,17,19,11, 9,27,25)( 4,18,20,12,10,28,26)( 5,14,29,22,23,16, 7, 6,13,30,21,24,15, 8)(31,32)$
14B5 $14,7^{2},2,1^{2}$ $16$ $14$ $26$ $( 3,11,25,19,27,17, 9)( 4,12,26,20,28,18,10)( 5,22, 7,30,15,14,23, 6,21, 8,29,16,13,24)(31,32)$
14B-5 $14,7^{2},2,1^{2}$ $16$ $14$ $26$ $( 3, 9,17,27,19,25,11)( 4,10,18,28,20,26,12)( 5,24,13,16,29, 8,21, 6,23,14,15,30, 7,22)(31,32)$
14C1 $14^{2},2^{2}$ $16$ $14$ $28$ $( 1,32)( 2,31)( 3,30, 9, 8,17,22,27, 6,19,24,25,14,11,16)( 4,29,10, 7,18,21,28, 5,20,23,26,13,12,15)$
14C-1 $14^{2},2^{2}$ $16$ $14$ $28$ $( 1,32)( 2,31)( 3,16,11,14,25,24,19, 6,27,22,17, 8, 9,30)( 4,15,12,13,26,23,20, 5,28,21,18, 7,10,29)$
14C3 $14^{2},2^{2}$ $16$ $14$ $28$ $( 1,32)( 2,31)( 3, 8,27,24,11,30,17, 6,25,16, 9,22,19,14)( 4, 7,28,23,12,29,18, 5,26,15,10,21,20,13)$
14C-3 $14^{2},2^{2}$ $16$ $14$ $28$ $( 1,32)( 2,31)( 3,14,19,22, 9,16,25, 6,17,30,11,24,27, 8)( 4,13,20,21,10,15,26, 5,18,29,12,23,28, 7)$
14C5 $14^{2},2^{2}$ $16$ $14$ $28$ $( 1,32)( 2,31)( 3,22,25,30,27,14, 9, 6,11, 8,19,16,17,24)( 4,21,26,29,28,13,10, 5,12, 7,20,15,18,23)$
14C-5 $14^{2},2^{2}$ $16$ $14$ $28$ $( 1,32)( 2,31)( 3,24,17,16,19, 8,11, 6, 9,14,27,30,25,22)( 4,23,18,15,20, 7,12, 5,10,13,28,29,26,21)$
28A1 $28,4$ $16$ $28$ $30$ $( 1,32, 2,31)( 3,30,10, 7,17,22,28, 5,19,24,26,13,11,16, 4,29, 9, 8,18,21,27, 6,20,23,25,14,12,15)$
28A-1 $28,4$ $16$ $28$ $30$ $( 1,32, 2,31)( 3,16,12,13,25,24,20, 5,27,22,18, 7, 9,30, 4,15,11,14,26,23,19, 6,28,21,17, 8,10,29)$
28A3 $28,4$ $16$ $28$ $30$ $( 1,32, 2,31)( 3, 8,28,23,11,30,18, 5,25,16,10,21,19,14, 4, 7,27,24,12,29,17, 6,26,15, 9,22,20,13)$
28A-3 $28,4$ $16$ $28$ $30$ $( 1,32, 2,31)( 3,14,20,21, 9,16,26, 5,17,30,12,23,27, 8, 4,13,19,22,10,15,25, 6,18,29,11,24,28, 7)$
28A5 $28,4$ $16$ $28$ $30$ $( 1,32, 2,31)( 3,22,26,29,27,14,10, 5,11, 8,20,15,17,24, 4,21,25,30,28,13, 9, 6,12, 7,19,16,18,23)$
28A-5 $28,4$ $16$ $28$ $30$ $( 1,32, 2,31)( 3,24,18,15,19, 8,12, 5, 9,14,28,29,25,22, 4,23,17,16,20, 7,11, 6,10,13,27,30,26,21)$

Malle's constant $a(G)$:     $1/8$

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Character table

40 x 40 character table

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Regular extensions

Data not computed