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Properties

Label 32T2852
32T2852 1 11 1->11 17 1->17 2 12 2->12 18 2->18 3 10 3->10 20 3->20 4 9 4->9 19 4->19 5 13 5->13 23 5->23 6 14 6->14 24 6->24 7 15 7->15 21 7->21 8 16 8->16 22 8->22 25 9->25 30 9->30 26 10->26 29 10->29 28 11->28 31 11->31 27 12->27 32 12->32 13->28 13->30 14->27 14->29 15->25 15->31 16->26 16->32 17->3 17->6 18->4 18->5 19->2 19->8 20->1 20->7 21->2 21->6 22->1 22->5 23->3 23->8 24->4 24->7 25->14 25->18 26->13 26->17 27->15 27->19 28->16 28->20 29->11 29->21 30->12 30->22 31->10 31->24 32->9 32->23
Degree $32$
Order $256$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_8^2:C_4$

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Copy content magma:G := TransitiveGroup(32, 2852);
 

Group invariants

Abstract group:  $C_8^2:C_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $256=2^{8}$
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:  $3$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $32$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $2852$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $8$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,17,6,24,4,19,8,22)(2,18,5,23,3,20,7,21)(9,25,14,29,11,28,16,32)(10,26,13,30,12,27,15,31)$, $(1,11,31,24,7,15,25,18,4,9,30,22,5,13,28,20)(2,12,32,23,8,16,26,17,3,10,29,21,6,14,27,19)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 6, $C_2^2$
$8$:  $D_{4}$ x 3, $C_4\times C_2$ x 3, $Q_8$
$16$:  $C_2^2:C_4$ x 3, $C_4^2$, $C_4:C_4$ x 3
$32$:  $C_4\wr C_2$ x 2, 32T41
$64$:  16T74, 16T90, 16T121
$128$:  32T1094

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 8: $C_2^2:C_4$, $C_4\wr C_2$ x 2

Degree 16: 16T121, 16T580 x 2

Low degree siblings

16T580 x 4, 32T2851 x 2, 32T2852, 32T2853 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 $1^{32}$ $1$ $1$ $0$ $()$
2A $2^{16}$ $1$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27)(29,32)(30,31)$
2B $2^{8},1^{16}$ $2$ $2$ $8$ $( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)$
2C $2^{16}$ $4$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)$
4A1 $4^{8}$ $1$ $4$ $24$ $( 1, 7, 4, 5)( 2, 8, 3, 6)( 9,13,11,15)(10,14,12,16)(17,21,19,23)(18,22,20,24)(25,30,28,31)(26,29,27,32)$
4A-1 $4^{8}$ $1$ $4$ $24$ $( 1, 5, 4, 7)( 2, 6, 3, 8)( 9,15,11,13)(10,16,12,14)(17,23,19,21)(18,24,20,22)(25,31,28,30)(26,32,27,29)$
4B $4^{8}$ $2$ $4$ $24$ $( 1, 7, 4, 5)( 2, 8, 3, 6)( 9,15,11,13)(10,16,12,14)(17,23,19,21)(18,24,20,22)(25,30,28,31)(26,29,27,32)$
4C1 $4^{4},2^{8}$ $2$ $4$ $20$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,15,11,13)(10,16,12,14)(17,23,19,21)(18,24,20,22)(25,28)(26,27)(29,32)(30,31)$
4C-1 $4^{4},2^{8}$ $2$ $4$ $20$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,13,11,15)(10,14,12,16)(17,21,19,23)(18,22,20,24)(25,28)(26,27)(29,32)(30,31)$
4D1 $4^{4},1^{16}$ $2$ $4$ $12$ $( 9,13,11,15)(10,14,12,16)(17,21,19,23)(18,22,20,24)$
4D-1 $4^{4},1^{16}$ $2$ $4$ $12$ $( 9,15,11,13)(10,16,12,14)(17,23,19,21)(18,24,20,22)$
4E $4^{8}$ $4$ $4$ $24$ $( 1, 6, 4, 8)( 2, 5, 3, 7)( 9,14,11,16)(10,13,12,15)(17,24,19,22)(18,23,20,21)(25,29,28,32)(26,30,27,31)$
4F $4^{4},2^{8}$ $8$ $4$ $20$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,16,11,14)(10,15,12,13)(17,22,19,24)(18,21,20,23)(25,27)(26,28)(29,31)(30,32)$
4G1 $4^{8}$ $16$ $4$ $24$ $( 1,21, 3,24)( 2,22, 4,23)( 5,17, 8,20)( 6,18, 7,19)( 9,28,10,27)(11,25,12,26)(13,31,14,32)(15,30,16,29)$
4G-1 $4^{8}$ $16$ $4$ $24$ $( 1,24, 3,21)( 2,23, 4,22)( 5,20, 8,17)( 6,19, 7,18)( 9,27,10,28)(11,26,12,25)(13,32,14,31)(15,29,16,30)$
8A1 $8^{2},4^{4}$ $4$ $8$ $26$ $( 1, 8, 4, 6)( 2, 7, 3, 5)( 9,22,15,18,11,24,13,20)(10,21,16,17,12,23,14,19)(25,29,28,32)(26,30,27,31)$
8A-1 $8^{2},4^{4}$ $4$ $8$ $26$ $( 1, 6, 4, 8)( 2, 5, 3, 7)( 9,18,13,22,11,20,15,24)(10,17,14,21,12,19,16,23)(25,32,28,29)(26,31,27,30)$
8B1 $8^{4}$ $4$ $8$ $28$ $( 1,27, 7,32, 4,26, 5,29)( 2,28, 8,31, 3,25, 6,30)( 9,21,15,17,11,23,13,19)(10,22,16,18,12,24,14,20)$
8B-1 $8^{4}$ $4$ $8$ $28$ $( 1,27, 7,32, 4,26, 5,29)( 2,28, 8,31, 3,25, 6,30)( 9,23,15,19,11,21,13,17)(10,24,16,20,12,22,14,18)$
8C1 $8^{2},4^{4}$ $4$ $8$ $26$ $( 1, 6, 4, 8)( 2, 5, 3, 7)( 9,24,15,20,11,22,13,18)(10,23,16,19,12,21,14,17)(25,32,28,29)(26,31,27,30)$
8C-1 $8^{2},4^{4}$ $4$ $8$ $26$ $( 1, 8, 4, 6)( 2, 7, 3, 5)( 9,20,13,24,11,18,15,22)(10,19,14,23,12,17,16,21)(25,29,28,32)(26,30,27,31)$
8D1 $8^{2},2^{8}$ $4$ $8$ $22$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,18,13,22,11,20,15,24)(10,17,14,21,12,19,16,23)(25,27)(26,28)(29,31)(30,32)$
8D-1 $8^{2},2^{8}$ $4$ $8$ $22$ $( 1,30, 5,25, 4,31, 7,28)( 2,29, 6,26, 3,32, 8,27)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)$
8E1 $8^{4}$ $4$ $8$ $28$ $( 1,26, 7,29, 4,27, 5,32)( 2,25, 8,30, 3,28, 6,31)( 9,17,13,21,11,19,15,23)(10,18,14,22,12,20,16,24)$
8E-1 $8^{4}$ $4$ $8$ $28$ $( 1,29, 5,26, 4,32, 7,27)( 2,30, 6,25, 3,31, 8,28)( 9,23,15,19,11,21,13,17)(10,24,16,20,12,22,14,18)$
8F1 $8^{2},2^{8}$ $4$ $8$ $22$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,24,15,20,11,22,13,18)(10,23,16,19,12,21,14,17)(25,26)(27,28)(29,30)(31,32)$
8F-1 $8^{2},2^{8}$ $4$ $8$ $22$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,20,13,24,11,18,15,22)(10,19,14,23,12,17,16,21)(25,26)(27,28)(29,30)(31,32)$
8G1 $8^{4}$ $4$ $8$ $28$ $( 1,28, 5,30, 4,25, 7,31)( 2,27, 6,29, 3,26, 8,32)( 9,20,15,22,11,18,13,24)(10,19,16,21,12,17,14,23)$
8G-1 $8^{4}$ $4$ $8$ $28$ $( 1,31, 7,25, 4,30, 5,28)( 2,32, 8,26, 3,29, 6,27)( 9,24,13,18,11,22,15,20)(10,23,14,17,12,21,16,19)$
8H $8^{4}$ $8$ $8$ $28$ $( 1,25, 5,31, 4,28, 7,30)( 2,26, 6,32, 3,27, 8,29)( 9,22,13,20,11,24,15,18)(10,21,14,19,12,23,16,17)$
8I1 $8^{2},4^{4}$ $8$ $8$ $26$ $( 1, 5, 4, 7)( 2, 6, 3, 8)( 9,21,13,19,11,23,15,17)(10,22,14,20,12,24,16,18)(25,30,28,31)(26,29,27,32)$
8I-1 $8^{2},4^{4}$ $8$ $8$ $26$ $( 1, 7, 4, 5)( 2, 8, 3, 6)( 9,17,15,23,11,19,13,21)(10,18,16,24,12,20,14,22)(25,31,28,30)(26,32,27,29)$
8J1 $8^{2},2^{4},1^{8}$ $8$ $8$ $18$ $( 9,17,15,23,11,19,13,21)(10,18,16,24,12,20,14,22)(25,28)(26,27)(29,32)(30,31)$
8J-1 $8^{2},2^{4},1^{8}$ $8$ $8$ $18$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,23,13,17,11,21,15,19)(10,24,14,18,12,22,16,20)$
8K1 $8^{4}$ $16$ $8$ $28$ $( 1,17, 6,24, 4,19, 8,22)( 2,18, 5,23, 3,20, 7,21)( 9,25,14,29,11,28,16,32)(10,26,13,30,12,27,15,31)$
8K-1 $8^{4}$ $16$ $8$ $28$ $( 1,20, 6,21, 4,18, 8,23)( 2,19, 5,22, 3,17, 7,24)( 9,26,14,30,11,27,16,31)(10,25,13,29,12,28,15,32)$
16A1 $16^{2}$ $16$ $16$ $30$ $( 1,16,28,21, 5,12,30,17, 4,14,25,23, 7,10,31,19)( 2,15,27,22, 6,11,29,18, 3,13,26,24, 8, 9,32,20)$
16A-1 $16^{2}$ $16$ $16$ $30$ $( 1, 9,30,24, 7,13,28,18, 4,11,31,22, 5,15,25,20)( 2,10,29,23, 8,14,27,17, 3,12,32,21, 6,16,26,19)$
16B1 $16^{2}$ $16$ $16$ $30$ $( 1,12,31,21, 7,16,25,19, 4,10,30,23, 5,14,28,17)( 2,11,32,22, 8,15,26,20, 3, 9,29,24, 6,13,27,18)$
16B-1 $16^{2}$ $16$ $16$ $30$ $( 1,13,25,24, 5, 9,31,20, 4,15,28,22, 7,11,30,18)( 2,14,26,23, 6,10,32,19, 3,16,27,21, 8,12,29,17)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

40 x 40 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed