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Properties

Label 34T25
Degree $34$
Order $4624$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{17}^2:\OD_{16}$

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Show commands: Magma

magma: G := TransitiveGroup(34, 25);
 

Group invariants

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

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$16$:  $C_8:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: None

Low degree siblings

34T26 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{34}$ $1$ $1$ $0$ $()$
2A $2^{8},1^{18}$ $34$ $2$ $8$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)$
2B $2^{16},1^{2}$ $289$ $2$ $16$ $( 1, 9)( 2, 8)( 3, 7)( 4, 6)(10,17)(11,16)(12,15)(13,14)(18,21)(19,20)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)$
4A1 $4^{8},1^{2}$ $289$ $4$ $24$ $( 1, 6, 9, 4)( 2,10, 8,17)( 3,14, 7,13)(11,12,16,15)(18,22,21,34)(19,26,20,30)(23,25,33,31)(24,29,32,27)$
4A-1 $4^{8},1^{2}$ $289$ $4$ $24$ $( 1, 4, 9, 6)( 2,17, 8,10)( 3,13, 7,14)(11,15,16,12)(18,34,21,22)(19,30,20,26)(23,31,33,25)(24,27,32,29)$
4B $4^{8},1^{2}$ $578$ $4$ $24$ $( 1,10, 8,16)( 2, 6, 7, 3)( 4,15, 5,11)( 9,12,17,14)(18,34,30,31)(19,21,29,27)(20,25,28,23)(22,33,26,32)$
8A1 $8^{4},2$ $578$ $8$ $29$ $( 1,22, 6,21, 9,34, 4,18)( 2,32,10,27, 8,24,17,29)( 3,25,14,33, 7,31,13,23)( 5,28)(11,20,12,30,16,19,15,26)$
8A-1 $8^{4},2$ $578$ $8$ $29$ $( 1,18, 4,34, 9,21, 6,22)( 2,29,17,24, 8,27,10,32)( 3,23,13,31, 7,33,14,25)( 5,28)(11,26,15,19,16,30,12,20)$
8B1 $8^{4},2$ $578$ $8$ $29$ $( 1,25,10,28,12,23, 3,20)( 2,31,14,18,11,34,16,30)( 4,26, 5,32, 9,22, 8,33)( 6,21,13,29, 7,27,17,19)(15,24)$
8B-1 $8^{4},2$ $578$ $8$ $29$ $( 1,20, 3,23,12,28,10,25)( 2,30,16,34,11,18,14,31)( 4,33, 8,22, 9,32, 5,26)( 6,19,17,27, 7,29,13,21)(15,24)$
17A1 $17,1^{17}$ $8$ $17$ $16$ $(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$
17A2 $17,1^{17}$ $8$ $17$ $16$ $(18,30,25,20,32,27,22,34,29,24,19,31,26,21,33,28,23)$
17A3 $17,1^{17}$ $8$ $17$ $16$ $(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$
17A6 $17,1^{17}$ $8$ $17$ $16$ $(18,20,22,24,26,28,30,32,34,19,21,23,25,27,29,31,33)$
17B1 $17^{2}$ $16$ $17$ $32$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$
17B3 $17^{2}$ $16$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$
17C1 $17^{2}$ $16$ $17$ $32$ $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$
17C3 $17^{2}$ $16$ $17$ $32$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$
17D1 $17^{2}$ $16$ $17$ $32$ $( 1,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$
17D2 $17^{2}$ $16$ $17$ $32$ $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$
17D3 $17^{2}$ $16$ $17$ $32$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$
17D6 $17^{2}$ $16$ $17$ $32$ $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$
17E1 $17^{2}$ $16$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,32,29,26,23,20,34,31,28,25,22,19,33,30,27,24,21)$
17E2 $17^{2}$ $16$ $17$ $32$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)$
17E3 $17^{2}$ $16$ $17$ $32$ $( 1,12, 6,17,11, 5,16,10, 4,15, 9, 3,14, 8, 2,13, 7)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$
17E6 $17^{2}$ $16$ $17$ $32$ $( 1,16,14,12,10, 8, 6, 4, 2,17,15,13,11, 9, 7, 5, 3)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$
17F1 $17^{2}$ $16$ $17$ $32$ $( 1, 6,11,16, 4, 9,14, 2, 7,12,17, 5,10,15, 3, 8,13)(18,29,23,34,28,22,33,27,21,32,26,20,31,25,19,30,24)$
17F2 $17^{2}$ $16$ $17$ $32$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$
17F3 $17^{2}$ $16$ $17$ $32$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$
17F6 $17^{2}$ $16$ $17$ $32$ $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$
34A1 $17,2^{8},1$ $136$ $34$ $24$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$
34A3 $17,2^{8},1$ $136$ $34$ $24$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$
34A7 $17,2^{8},1$ $136$ $34$ $24$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$
34A9 $17,2^{8},1$ $136$ $34$ $24$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$

Malle's constant $a(G)$:     $1/8$

magma: ConjugacyClasses(G);
 

Character table

34 x 34 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed