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Properties

Label 34T28
Degree $34$
Order $8160$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $\SOMinus(4,4)$

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Show commands: Magma

magma: G := TransitiveGroup(34, 28);
 

Group invariants

Abstract group:  $\SOMinus(4,4)$
magma: IdentifyGroup(G);
 
Order:  $8160=2^{5} \cdot 3 \cdot 5 \cdot 17$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
magma: NilpotencyClass(G);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 17: $\PSL(2,16):C_2$

Low degree siblings

17T7

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 $2^{17}$ $68$ $2$ $17$ $( 1,26)( 2,25)( 3,13)( 4,14)( 5, 7)( 6, 8)( 9,10)(11,12)(15,16)(17,33)(18,34)(19,22)(20,21)(23,24)(27,28)(29,31)(30,32)$
2B $2^{16},1^{2}$ $255$ $2$ $16$ $( 1,14)( 2,13)( 3,25)( 4,26)( 5, 8)( 6, 7)( 9,27)(10,28)(15,23)(16,24)(17,31)(18,32)(19,21)(20,22)(29,33)(30,34)$
3A $3^{10},1^{4}$ $272$ $3$ $20$ $( 1,24,33)( 2,23,34)( 3,16,32)( 4,15,31)( 5,19, 8)( 6,20, 7)( 9,30,26)(10,29,25)(13,27,17)(14,28,18)$
4A $4^{8},2$ $1020$ $4$ $25$ $( 1,34,14,30)( 2,33,13,29)( 3,17,25,31)( 4,18,26,32)( 5,24, 8,16)( 6,23, 7,15)( 9,20,27,22)(10,19,28,21)(11,12)$
5A1 $5^{6},1^{4}$ $272$ $5$ $24$ $( 1, 7, 3,10,28)( 2, 8, 4, 9,27)( 5,15,30,17,23)( 6,16,29,18,24)(13,34,19,31,26)(14,33,20,32,25)$
5A2 $5^{6},1^{4}$ $272$ $5$ $24$ $( 1, 3,28, 7,10)( 2, 4,27, 8, 9)( 5,30,23,15,17)( 6,29,24,16,18)(13,19,26,34,31)(14,20,25,33,32)$
6A $6^{5},2^{2}$ $1360$ $6$ $27$ $( 1, 8,22,26, 6,19)( 2, 7,21,25, 5,20)( 3,17,29,13,33,31)( 4,18,30,14,34,32)( 9,10)(11,28,23,12,27,24)(15,16)$
10A1 $10^{3},2^{2}$ $816$ $10$ $29$ $( 1, 8,25,13,22,34,20, 9,28,11)( 2, 7,26,14,21,33,19,10,27,12)( 3,31)( 4,32)( 5,18,23,29,15, 6,17,24,30,16)$
10A3 $10^{3},2^{2}$ $816$ $10$ $29$ $( 1,13,20,11,25,34,28, 8,22, 9)( 2,14,19,12,26,33,27, 7,21,10)( 3,31)( 4,32)( 5,29,17,16,23, 6,30,18,15,24)$
15A1 $15^{2},1^{4}$ $544$ $15$ $28$ $( 1,32,18, 7,25,24, 3,14, 6,10,33,16,28,20,29)( 2,31,17, 8,26,23, 4,13, 5, 9,34,15,27,19,30)$
15A2 $15^{2},1^{4}$ $544$ $15$ $28$ $( 1,18,25, 3, 6,33,28,29,32, 7,24,14,10,16,20)( 2,17,26, 4, 5,34,27,30,31, 8,23,13, 9,15,19)$
17A1 $17^{2}$ $480$ $17$ $32$ $( 1,24,14,16, 6,28,29,25, 7,20,12,33,18,10,22, 3,32)( 2,23,13,15, 5,27,30,26, 8,19,11,34,17, 9,21, 4,31)$
17A2 $17^{2}$ $480$ $17$ $32$ $( 1,14, 6,29, 7,12,18,22,32,24,16,28,25,20,33,10, 3)( 2,13, 5,30, 8,11,17,21,31,23,15,27,26,19,34, 9, 4)$
17A3 $17^{2}$ $480$ $17$ $32$ $( 1,16,29,20,18, 3,24, 6,25,12,10,32,14,28, 7,33,22)( 2,15,30,19,17, 4,23, 5,26,11, 9,31,13,27, 8,34,21)$
17A6 $17^{2}$ $480$ $17$ $32$ $( 1,29,18,24,25,10,14, 7,22,16,20, 3, 6,12,32,28,33)( 2,30,17,23,26, 9,13, 8,21,15,19, 4, 5,11,31,27,34)$

Malle's constant $a(G)$:     $1/16$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 2B 3A 4A 5A1 5A2 6A 10A1 10A3 15A1 15A2 17A1 17A2 17A3 17A6
Size 1 68 255 272 1020 272 272 1360 816 816 544 544 480 480 480 480
2 P 1A 1A 1A 3A 2B 5A2 5A1 3A 5A1 5A2 15A2 15A1 17A2 17A1 17A6 17A3
3 P 1A 2A 2B 1A 4A 5A2 5A1 2A 10A3 10A1 5A1 5A2 17A3 17A6 17A2 17A1
5 P 1A 2A 2B 3A 4A 1A 1A 6A 2A 2A 3A 3A 17A3 17A6 17A2 17A1
17 P 1A 2A 2B 3A 4A 5A2 5A1 6A 10A3 10A1 15A2 15A1 1A 1A 1A 1A
Type
8160.a.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
8160.a.1b R 1 −1 1 1 −1 1 1 −1 −1 −1 1 1 1 1 1 1
8160.a.16a R 16 4 0 1 0 1 1 1 −1 −1 1 1 −1 −1 −1 −1
8160.a.16b R 16 −4 0 1 0 1 1 −1 1 1 1 1 −1 −1 −1 −1
8160.a.17a R 17 5 1 −1 1 2 2 −1 0 0 −1 −1 0 0 0 0
8160.a.17b R 17 −5 1 −1 −1 2 2 1 0 0 −1 −1 0 0 0 0
8160.a.17c1 R 17 3 1 2 −1 ζ5−2+ζ52 ζ5−1+ζ5 0 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−1+ζ5 ζ5−2+ζ52 0 0 0 0
8160.a.17c2 R 17 3 1 2 −1 ζ5−1+ζ5 ζ5−2+ζ52 0 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2+ζ52 ζ5−1+ζ5 0 0 0 0
8160.a.17d1 R 17 −3 1 2 1 ζ5−2+ζ52 ζ5−1+ζ5 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 0 0 0 0
8160.a.17d2 R 17 −3 1 2 1 ζ5−1+ζ5 ζ5−2+ζ52 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 0 0 0 0
8160.a.30a1 R 30 0 −2 0 0 0 0 0 0 0 0 0 −ζ17−4−ζ17−1−ζ17−ζ174 −ζ17−8−ζ17−2−ζ172−ζ178 −ζ17−5−ζ17−3−ζ173−ζ175 −ζ17−7−ζ17−6−ζ176−ζ177
8160.a.30a2 R 30 0 −2 0 0 0 0 0 0 0 0 0 −ζ17−5−ζ17−3−ζ173−ζ175 −ζ17−7−ζ17−6−ζ176−ζ177 −ζ17−8−ζ17−2−ζ172−ζ178 −ζ17−4−ζ17−1−ζ17−ζ174
8160.a.30a3 R 30 0 −2 0 0 0 0 0 0 0 0 0 −ζ17−8−ζ17−2−ζ172−ζ178 −ζ17−4−ζ17−1−ζ17−ζ174 −ζ17−7−ζ17−6−ζ176−ζ177 −ζ17−5−ζ17−3−ζ173−ζ175
8160.a.30a4 R 30 0 −2 0 0 0 0 0 0 0 0 0 −ζ17−7−ζ17−6−ζ176−ζ177 −ζ17−5−ζ17−3−ζ173−ζ175 −ζ17−4−ζ17−1−ζ17−ζ174 −ζ17−8−ζ17−2−ζ172−ζ178
8160.a.34a1 R 34 0 2 −2 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 0 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 0 0
8160.a.34a2 R 34 0 2 −2 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 0 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 0 0

magma: CharacterTable(G);
 

Regular extensions

Data not computed