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Properties

Label 34T37
Degree $34$
Order $9248$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{17}^2:\SD_{32}$

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Show commands: Magma

magma: G := TransitiveGroup(34, 37);
 

Group invariants

Abstract group:  $C_{17}^2:\SD_{32}$
magma: IdentifyGroup(G);
 
Order:  $9248=2^{5} \cdot 17^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $34$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $37$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,32,12,22,6,29,17,19,11,26,5,33,16,23,10,30,4,20,15,27,9,34,3,24,14,31,8,21,2,28,13,18,7,25)$, $(1,32,12,34)(2,26,11,23)(3,20,10,29)(4,31,9,18)(5,25,8,24)(6,19,7,30)(13,28,17,21)(14,22,16,27)(15,33)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$
$16$:  $D_{8}$
$32$:  16T55

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: None

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{34}$ $1$ $1$ $0$ $()$
2A $2^{17}$ $136$ $2$ $17$ $( 1,28)( 2,26)( 3,24)( 4,22)( 5,20)( 6,18)( 7,33)( 8,31)( 9,29)(10,27)(11,25)(12,23)(13,21)(14,19)(15,34)(16,32)(17,30)$
2B $2^{16},1^{2}$ $289$ $2$ $16$ $( 1,11)( 2,10)( 3, 9)( 4, 8)( 5, 7)(12,17)(13,16)(14,15)(18,22)(19,21)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)$
4A $4^{8},1^{2}$ $578$ $4$ $24$ $( 1, 3,11, 9)( 2, 7,10, 5)( 4,15, 8,14)(12,13,17,16)(18,28,22,29)(19,24,21,33)(23,25,34,32)(26,30,31,27)$
4B $4^{8},2$ $2312$ $4$ $25$ $( 1,19, 9,24)( 2,26, 8,34)( 3,33, 7,27)( 4,23, 6,20)( 5,30)(10,31,17,29)(11,21,16,22)(12,28,15,32)(13,18,14,25)$
8A1 $8^{4},1^{2}$ $578$ $8$ $28$ $( 1,16, 3,12,11,13, 9,17)( 2,14, 7, 4,10,15, 5, 8)(18,21,28,33,22,19,29,24)(23,27,25,26,34,30,32,31)$
8A3 $8^{4},1^{2}$ $578$ $8$ $28$ $( 1,12, 9,16,11,17, 3,13)( 2, 4, 5,14,10, 8, 7,15)(18,33,29,21,22,24,28,19)(23,26,32,27,34,31,25,30)$
16A1 $16^{2},1^{2}$ $578$ $16$ $30$ $( 1, 5,16, 8, 3, 2,12,14,11, 7,13, 4, 9,10,17,15)(18,30,21,32,28,31,33,23,22,27,19,25,29,26,24,34)$
16A-1 $16^{2},1^{2}$ $578$ $16$ $30$ $( 1,15,17,10, 9, 4,13, 7,11,14,12, 2, 3, 8,16, 5)(18,34,24,26,29,25,19,27,22,23,33,31,28,32,21,30)$
16A3 $16^{2},1^{2}$ $578$ $16$ $30$ $( 1, 8,12, 7, 9,15,16, 2,11, 4,17, 5, 3,14,13,10)(18,32,33,27,29,34,21,31,22,25,24,30,28,23,19,26)$
16A-3 $16^{2},1^{2}$ $578$ $16$ $30$ $( 1,10,13,14, 3, 5,17, 4,11, 2,16,15, 9, 7,12, 8)(18,26,19,23,28,30,24,25,22,31,21,34,29,27,33,32)$
17A1 $17^{2}$ $16$ $17$ $32$ $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$
17A2 $17^{2}$ $16$ $17$ $32$ $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$
17A3 $17^{2}$ $16$ $17$ $32$ $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$
17A4 $17^{2}$ $16$ $17$ $32$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$
17A5 $17^{2}$ $16$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$
17A6 $17^{2}$ $16$ $17$ $32$ $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$
17A7 $17^{2}$ $16$ $17$ $32$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$
17A8 $17^{2}$ $16$ $17$ $32$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$
17B $17,1^{17}$ $32$ $17$ $16$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)$
17C1 $17^{2}$ $32$ $17$ $32$ $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$
17C2 $17^{2}$ $32$ $17$ $32$ $( 1,14,10, 6, 2,15,11, 7, 3,16,12, 8, 4,17,13, 9, 5)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$
17C3 $17^{2}$ $32$ $17$ $32$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$
17C6 $17^{2}$ $32$ $17$ $32$ $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$
34A1 $34$ $272$ $34$ $33$ $( 1,30,16,34,14,21,12,25,10,29, 8,33, 6,20, 4,24, 2,28,17,32,15,19,13,23,11,27, 9,31, 7,18, 5,22, 3,26)$
34A3 $34$ $272$ $34$ $33$ $( 1,34,12,29, 6,24,17,19,11,31, 5,26,16,21,10,33, 4,28,15,23, 9,18, 3,30,14,25, 8,20, 2,32,13,27, 7,22)$
34A5 $34$ $272$ $34$ $33$ $( 1,21, 8,24,15,27, 5,30,12,33, 2,19, 9,22,16,25, 6,28,13,31, 3,34,10,20,17,23, 7,26,14,29, 4,32,11,18)$
34A7 $34$ $272$ $34$ $33$ $( 1,25, 4,19, 7,30,10,24,13,18,16,29, 2,23, 5,34, 8,28,11,22,14,33,17,27, 3,21, 6,32, 9,26,12,20,15,31)$
34A9 $34$ $272$ $34$ $33$ $( 1,29,17,31,16,33,15,18,14,20,13,22,12,24,11,26,10,28, 9,30, 8,32, 7,34, 6,19, 5,21, 4,23, 3,25, 2,27)$
34A11 $34$ $272$ $34$ $33$ $( 1,33,13,26, 8,19, 3,29,15,22,10,32, 5,25,17,18,12,28, 7,21, 2,31,14,24, 9,34, 4,27,16,20,11,30, 6,23)$
34A13 $34$ $272$ $34$ $33$ $( 1,20, 9,21,17,22, 8,23,16,24, 7,25,15,26, 6,27,14,28, 5,29,13,30, 4,31,12,32, 3,33,11,34, 2,18,10,19)$
34A15 $34$ $272$ $34$ $33$ $( 1,24, 5,33, 9,25,13,34,17,26, 4,18, 8,27,12,19,16,28, 3,20, 7,29,11,21,15,30, 2,22, 6,31,10,23,14,32)$

Malle's constant $a(G)$:     $1/16$

magma: ConjugacyClasses(G);
 

Character table

32 x 32 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed