Properties

Label 34T23
Degree $34$
Order $4624$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{17}^2:\SD_{16}$

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Show commands: Magma

magma: G := TransitiveGroup(34, 23);
 

Group invariants

Abstract group:  $C_{17}^2:\SD_{16}$
magma: IdentifyGroup(G);
 
Order:  $4624=2^{4} \cdot 17^{2}$
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 $t$:  $23$
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:  $(1,22,11,30,4,21,14,29,7,20,17,28,10,19,3,27,13,18,6,26,16,34,9,25,2,33,12,24,5,32,15,23,8,31)$, $(1,5,13,12,10,6,15,16)(2,7,17,3,9,4,11,8)(18,24,21,31,26,20,23,30)(19,32,34,33,25,29,27,28)$
magma: Generators(G);
 

Low degree resolvents

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

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 $2^{17}$ $68$ $2$ $17$ $( 1,27)( 2,21)( 3,32)( 4,26)( 5,20)( 6,31)( 7,25)( 8,19)( 9,30)(10,24)(11,18)(12,29)(13,23)(14,34)(15,28)(16,22)(17,33)$
2B $2^{16},1^{2}$ $289$ $2$ $16$ $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,17)(14,16)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)$
4A $4^{8},1^{2}$ $578$ $4$ $24$ $( 1,10,12, 3)( 2,14,11,16)( 4, 5, 9, 8)( 6,13, 7,17)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)$
4B $4^{8},2$ $1156$ $4$ $25$ $( 1,21,15,19)( 2,33,14,24)( 3,28,13,29)( 4,23,12,34)( 5,18,11,22)( 6,30,10,27)( 7,25, 9,32)( 8,20)(16,31,17,26)$
8A1 $8^{4},1^{2}$ $578$ $8$ $28$ $( 1, 9,10, 8,12, 4, 3, 5)( 2, 7,14,17,11, 6,16,13)(19,27,31,33,34,26,22,20)(21,28,23,29,32,25,30,24)$
8A-1 $8^{4},1^{2}$ $578$ $8$ $28$ $( 1, 4,10, 5,12, 9, 3, 8)( 2, 6,14,13,11, 7,16,17)(19,26,31,20,34,27,22,33)(21,25,23,24,32,28,30,29)$
17A1 $17^{2}$ $8$ $17$ $32$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$
17A2 $17^{2}$ $8$ $17$ $32$ $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$
17A3 $17^{2}$ $8$ $17$ $32$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$
17A4 $17^{2}$ $8$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$
17A5 $17^{2}$ $8$ $17$ $32$ $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$
17A6 $17^{2}$ $8$ $17$ $32$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$
17A7 $17^{2}$ $8$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$
17A8 $17^{2}$ $8$ $17$ $32$ $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$
17B1 $17^{2}$ $16$ $17$ $32$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,20,22,24,26,28,30,32,34,19,21,23,25,27,29,31,33)$
17B3 $17^{2}$ $16$ $17$ $32$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$
17C1 $17,1^{17}$ $16$ $17$ $16$ $(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$
17C2 $17^{2}$ $16$ $17$ $32$ $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19)$
17C3 $17^{2}$ $16$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$
17C6 $17^{2}$ $16$ $17$ $32$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$
17D1 $17,1^{17}$ $16$ $17$ $16$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)$
17D2 $17^{2}$ $16$ $17$ $32$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$
17D3 $17^{2}$ $16$ $17$ $32$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$
17D4 $17^{2}$ $16$ $17$ $32$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$
17D5 $17^{2}$ $16$ $17$ $32$ $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$
17D6 $17^{2}$ $16$ $17$ $32$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$
17D7 $17^{2}$ $16$ $17$ $32$ $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$
17D8 $17^{2}$ $16$ $17$ $32$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$
34A1 $34$ $136$ $34$ $33$ $( 1,22,16,34,14,29,12,24,10,19, 8,31, 6,26, 4,21, 2,33,17,28,15,23,13,18,11,30, 9,25, 7,20, 5,32, 3,27)$
34A3 $34$ $136$ $34$ $33$ $( 1,26, 4,25, 7,24,10,23,13,22,16,21, 2,20, 5,19, 8,18,11,34,14,33,17,32, 3,31, 6,30, 9,29,12,28,15,27)$
34A5 $34$ $136$ $34$ $33$ $( 1,20, 5,30, 9,23,13,33,17,26, 4,19, 8,29,12,22,16,32, 3,25, 7,18,11,28,15,21, 2,31, 6,24,10,34,14,27)$
34A7 $34$ $136$ $34$ $33$ $( 1,24,10,21, 2,18,11,32, 3,29,12,26, 4,23,13,20, 5,34,14,31, 6,28,15,25, 7,22,16,19, 8,33,17,30, 9,27)$
34A9 $34$ $136$ $34$ $33$ $( 1,21, 2,32, 3,26, 4,20, 5,31, 6,25, 7,19, 8,30, 9,24,10,18,11,29,12,23,13,34,14,28,15,22,16,33,17,27)$
34A11 $34$ $136$ $34$ $33$ $( 1,25, 7,23,13,21, 2,19, 8,34,14,32, 3,30, 9,28,15,26, 4,24,10,22,16,20, 5,18,11,33,17,31, 6,29,12,27)$
34A13 $34$ $136$ $34$ $33$ $( 1,18,11,26, 4,34,14,25, 7,33,17,24,10,32, 3,23,13,31, 6,22,16,30, 9,21, 2,29,12,20, 5,28,15,19, 8,27)$
34A15 $34$ $136$ $34$ $33$ $( 1,23,13,19, 8,32, 3,28,15,24,10,20, 5,33,17,29,12,25, 7,21, 2,34,14,30, 9,26, 4,22,16,18,11,31, 6,27)$

Malle's constant $a(G)$:     $1/16$

magma: ConjugacyClasses(G);
 

Character table

37 x 37 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed