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Properties

Label 32T2853
32T2853 1 12 1->12 22 1->22 2 11 2->11 21 2->21 3 10 3->10 24 3->24 4 9 4->9 23 4->23 5 13 5->13 17 5->17 6 14 6->14 18 6->18 7 15 7->15 19 7->19 8 16 8->16 20 8->20 9->1 9->3 10->2 10->4 11->1 11->3 12->2 12->4 13->6 13->8 14->5 14->7 15->6 15->8 16->5 16->7 25 17->25 28 17->28 26 18->26 27 18->27 19->25 19->28 20->26 20->27 30 21->30 31 21->31 29 22->29 32 22->32 23->30 23->31 24->29 24->32 25->9 25->18 26->10 26->17 27->12 27->19 28->11 28->20 29->14 29->21 30->13 30->22 31->15 31->24 32->16 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, 2853);
 

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 $t$:  $2853$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $16$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,12,2,11)(3,10,4,9)(5,13,6,14)(7,15,8,16)(17,28,20,26)(18,27,19,25)(21,31,24,29)(22,32,23,30)$, $(1,22,29,14,7,19,28,11,3,24,32,16,5,17,25,9)(2,21,30,13,8,20,27,12,4,23,31,15,6,18,26,10)$
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

Low degree siblings

16T580 x 4, 32T2851 x 2, 32T2852 x 2, 32T2853 x 3

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

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