Properties

Label 34T40
Degree $34$
Order $9248$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{17}^2.D_4$

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Show commands: Magma

magma: G := TransitiveGroup(34, 40);
 

Group invariants

Abstract group:  $D_{17}^2.D_4$
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 $t$:  $40$
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:  $(1,13,8,3,15,10,5,17,12,7,2,14,9,4,16,11,6)(18,33,25,27)(19,20,24,23)(21,28,22,32)(26,31,34,29)$, $(1,18,10,21,2,24,11,27,3,30,12,33,4,19,13,22,5,25,14,28,6,31,15,34,7,20,16,23,8,26,17,29,9,32)$
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, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$32$:  $C_4\wr C_2$

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 $2^{8},1^{18}$ $34$ $2$ $8$ $(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)$
2B $2^{17}$ $68$ $2$ $17$ $( 1,25)( 2,31)( 3,20)( 4,26)( 5,32)( 6,21)( 7,27)( 8,33)( 9,22)(10,28)(11,34)(12,23)(13,29)(14,18)(15,24)(16,30)(17,19)$
2C $2^{16},1^{2}$ $289$ $2$ $16$ $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,17)(12,16)(13,15)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)$
4A1 $4^{4},1^{18}$ $34$ $4$ $12$ $(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)$
4A-1 $4^{4},1^{18}$ $34$ $4$ $12$ $(19,22,34,31)(20,26,33,27)(21,30,32,23)(24,25,29,28)$
4B1 $4^{8},1^{2}$ $289$ $4$ $24$ $( 1,13,10,15)( 2,17, 9,11)( 3, 4, 8, 7)( 5,12, 6,16)(19,22,34,31)(20,26,33,27)(21,30,32,23)(24,25,29,28)$
4B-1 $4^{8},1^{2}$ $289$ $4$ $24$ $( 1,15,10,13)( 2,11, 9,17)( 3, 7, 8, 4)( 5,16, 6,12)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)$
4C $4^{8},1^{2}$ $578$ $4$ $24$ $( 1,15,10,13)( 2,11, 9,17)( 3, 7, 8, 4)( 5,16, 6,12)(19,22,34,31)(20,26,33,27)(21,30,32,23)(24,25,29,28)$
4D1 $4^{4},2^{8},1^{2}$ $578$ $4$ $20$ $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,17)(12,16)(13,15)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)$
4D-1 $4^{4},2^{8},1^{2}$ $578$ $4$ $20$ $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,17)(12,16)(13,15)(19,22,34,31)(20,26,33,27)(21,30,32,23)(24,25,29,28)$
4E $4^{8},2$ $1156$ $4$ $25$ $( 1,25,10,28)( 2,31, 9,22)( 3,20, 8,33)( 4,26, 7,27)( 5,32, 6,21)(11,34,17,19)(12,23,16,30)(13,29,15,24)(14,18)$
8A1 $8^{4},2$ $1156$ $8$ $29$ $( 1,24,13,25,10,29,15,28)( 2,34,17,31, 9,19,11,22)( 3,27, 4,20, 8,26, 7,33)( 5,30,12,32, 6,23,16,21)(14,18)$
8A-1 $8^{4},2$ $1156$ $8$ $29$ $( 1,24,15,28,10,29,13,25)( 2,34,11,22, 9,19,17,31)( 3,27, 7,33, 8,26, 4,20)( 5,30,16,21, 6,23,12,32)(14,18)$
17A1 $17,1^{17}$ $8$ $17$ $16$ $(18,23,28,33,21,26,31,19,24,29,34,22,27,32,20,25,30)$
17A2 $17,1^{17}$ $8$ $17$ $16$ $(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$
17A3 $17,1^{17}$ $8$ $17$ $16$ $(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$
17A6 $17,1^{17}$ $8$ $17$ $16$ $(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$
17B1 $17^{2}$ $16$ $17$ $32$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$
17B2 $17^{2}$ $16$ $17$ $32$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,21,24,27,30,33,19,22,25,28,31,34,20,23,26,29,32)$
17B3 $17^{2}$ $16$ $17$ $32$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$
17B6 $17^{2}$ $16$ $17$ $32$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,27,19,28,20,29,21,30,22,31,23,32,24,33,25,34,26)$
17C1 $17^{2}$ $32$ $17$ $32$ $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)(18,28,21,31,24,34,27,20,30,23,33,26,19,29,22,32,25)$
17C3 $17^{2}$ $32$ $17$ $32$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$
17D1 $17^{2}$ $32$ $17$ $32$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,33,31,29,27,25,23,21,19,34,32,30,28,26,24,22,20)$
17D2 $17^{2}$ $32$ $17$ $32$ $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,24,30,19,25,31,20,26,32,21,27,33,22,28,34,23,29)$
17D3 $17^{2}$ $32$ $17$ $32$ $( 1,11, 4,14, 7,17,10, 3,13, 6,16, 9, 2,12, 5,15, 8)(18,31,27,23,19,32,28,24,20,33,29,25,21,34,30,26,22)$
17D6 $17^{2}$ $32$ $17$ $32$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,26,34,25,33,24,32,23,31,22,30,21,29,20,28,19,27)$
34A1 $17,2^{8},1$ $136$ $34$ $24$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,26)(19,25)(20,24)(21,23)(27,34)(28,33)(29,32)(30,31)$
34A3 $17,2^{8},1$ $136$ $34$ $24$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,25)(19,24)(20,23)(21,22)(26,34)(27,33)(28,32)(29,31)$
34A7 $17,2^{8},1$ $136$ $34$ $24$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,22)(19,21)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)$
34A9 $17,2^{8},1$ $136$ $34$ $24$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(33,34)$
34B1 $34$ $272$ $34$ $33$ $( 1,25,14,18,10,28, 6,21, 2,31,15,24,11,34, 7,27, 3,20,16,30,12,23, 8,33, 4,26,17,19,13,29, 9,22, 5,32)$
34B3 $34$ $272$ $34$ $33$ $( 1,25, 6,21,11,34,16,30, 4,26, 9,22,14,18, 2,31, 7,27,12,23,17,19, 5,32,10,28,15,24, 3,20, 8,33,13,29)$
34B7 $34$ $272$ $34$ $33$ $( 1,25,12,23, 6,21,17,19,11,34, 5,32,16,30,10,28, 4,26,15,24, 9,22, 3,20,14,18, 8,33, 2,31,13,29, 7,27)$
34B9 $34$ $272$ $34$ $33$ $( 1,25,16,30,14,18,12,23,10,28, 8,33, 6,21, 4,26, 2,31,17,19,15,24,13,29,11,34, 9,22, 7,27, 5,32, 3,20)$
68A1 $17,4^{4},1$ $136$ $68$ $28$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,24,31,25)(19,28,30,21)(20,32,29,34)(22,23,27,26)$
68A-1 $17,4^{4},1$ $136$ $68$ $28$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,30,27,32)(19,34,26,28)(20,21,25,24)(22,29,23,33)$
68A3 $17,4^{4},1$ $136$ $68$ $28$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,20,28,26)(19,24,27,22)(21,32,25,31)(29,30,34,33)$
68A-3 $17,4^{4},1$ $136$ $68$ $28$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,34,21,22)(19,30,20,26)(23,31,33,25)(24,27,32,29)$
68A7 $17,4^{4},1$ $136$ $68$ $28$ $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,19,23,22)(20,27,21,31)(24,26,34,32)(25,30,33,28)$
68A-7 $17,4^{4},1$ $136$ $68$ $28$ $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,23,20,32)(21,28,34,27)(22,24,33,31)(25,29,30,26)$
68A9 $17,4^{4},1$ $136$ $68$ $28$ $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,33,24,26)(19,29,23,30)(20,25,22,34)(27,31,32,28)$
68A-9 $17,4^{4},1$ $136$ $68$ $28$ $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,29,19,25)(20,21,34,33)(22,30,32,24)(23,26,31,28)$

Malle's constant $a(G)$:     $1/8$

magma: ConjugacyClasses(G);
 

Character table

44 x 44 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed