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Properties

Label 29T6
Degree $29$
Order $812$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $F_{29}$

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Show commands: Magma

magma: G := TransitiveGroup(29, 6);
 

Group action invariants

Degree $n$:  $29$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $6$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $F_{29}$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29), (1,2,3,7,4,24,8,14,5,12,25,27,9,20,15,29,6,23,13,11,26,19,28,22,10,18,21,17,16)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$7$:  $C_7$
$14$:  $C_{14}$
$28$:  $C_{28}$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - 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 TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 29 $ $28$ $29$ $( 1,14,15,19,16, 8,20,26,17,24, 9,11,21, 4,27,13,18, 7,25,23,10, 3,12, 6,22, 2, 5,29,28)$
$ 7, 7, 7, 7, 1 $ $29$ $7$ $( 2, 6,10,14,18,22,26)( 3, 7,11,15,19,23,27)( 4, 8,12,16,20,24,28) ( 5, 9,13,17,21,25,29)$
$ 7, 7, 7, 7, 1 $ $29$ $7$ $( 2,14,26,10,22, 6,18)( 3,15,27,11,23, 7,19)( 4,16,28,12,24, 8,20) ( 5,17,29,13,25, 9,21)$
$ 7, 7, 7, 7, 1 $ $29$ $7$ $( 2,10,18,26, 6,14,22)( 3,11,19,27, 7,15,23)( 4,12,20,28, 8,16,24) ( 5,13,21,29, 9,17,25)$
$ 7, 7, 7, 7, 1 $ $29$ $7$ $( 2,26,22,18,14,10, 6)( 3,27,23,19,15,11, 7)( 4,28,24,20,16,12, 8) ( 5,29,25,21,17,13, 9)$
$ 7, 7, 7, 7, 1 $ $29$ $7$ $( 2,18, 6,22,10,26,14)( 3,19, 7,23,11,27,15)( 4,20, 8,24,12,28,16) ( 5,21, 9,25,13,29,17)$
$ 7, 7, 7, 7, 1 $ $29$ $7$ $( 2,22,14, 6,26,18,10)( 3,23,15, 7,27,19,11)( 4,24,16, 8,28,20,12) ( 5,25,17, 9,29,21,13)$
$ 14, 14, 1 $ $29$ $14$ $( 2, 4, 6, 8,10,12,14,16,18,20,22,24,26,28)( 3, 5, 7, 9,11,13,15,17,19,21,23, 25,27,29)$
$ 14, 14, 1 $ $29$ $14$ $( 2, 8,14,20,26, 4,10,16,22,28, 6,12,18,24)( 3, 9,15,21,27, 5,11,17,23,29, 7, 13,19,25)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $29$ $2$ $( 2,16)( 3,17)( 4,18)( 5,19)( 6,20)( 7,21)( 8,22)( 9,23)(10,24)(11,25)(12,26) (13,27)(14,28)(15,29)$
$ 14, 14, 1 $ $29$ $14$ $( 2,12,22, 4,14,24, 6,16,26, 8,18,28,10,20)( 3,13,23, 5,15,25, 7,17,27, 9,19, 29,11,21)$
$ 14, 14, 1 $ $29$ $14$ $( 2,28,26,24,22,20,18,16,14,12,10, 8, 6, 4)( 3,29,27,25,23,21,19,17,15,13,11, 9, 7, 5)$
$ 14, 14, 1 $ $29$ $14$ $( 2,20,10,28,18, 8,26,16, 6,24,14, 4,22,12)( 3,21,11,29,19, 9,27,17, 7,25,15, 5,23,13)$
$ 14, 14, 1 $ $29$ $14$ $( 2,24,18,12, 6,28,22,16,10, 4,26,20,14, 8)( 3,25,19,13, 7,29,23,17,11, 5,27, 21,15, 9)$
$ 28, 1 $ $29$ $28$ $( 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26, 27,28,29)$
$ 28, 1 $ $29$ $28$ $( 2, 7,12,17,22,27, 4, 9,14,19,24,29, 6,11,16,21,26, 3, 8,13,18,23,28, 5,10, 15,20,25)$
$ 28, 1 $ $29$ $28$ $( 2,15,28,13,26,11,24, 9,22, 7,20, 5,18, 3,16,29,14,27,12,25,10,23, 8,21, 6, 19, 4,17)$
$ 28, 1 $ $29$ $28$ $( 2,11,20,29,10,19,28, 9,18,27, 8,17,26, 7,16,25, 6,15,24, 5,14,23, 4,13,22, 3,12,21)$
$ 28, 1 $ $29$ $28$ $( 2,27,24,21,18,15,12, 9, 6, 3,28,25,22,19,16,13,10, 7, 4,29,26,23,20,17,14, 11, 8, 5)$
$ 28, 1 $ $29$ $28$ $( 2,19, 8,25,14, 3,20, 9,26,15, 4,21,10,27,16, 5,22,11,28,17, 6,23,12,29,18, 7,24,13)$
$ 4, 4, 4, 4, 4, 4, 4, 1 $ $29$ $4$ $( 2,23,16, 9)( 3,24,17,10)( 4,25,18,11)( 5,26,19,12)( 6,27,20,13)( 7,28,21,14) ( 8,29,22,15)$
$ 28, 1 $ $29$ $28$ $( 2, 5, 8,11,14,17,20,23,26,29, 4, 7,10,13,16,19,22,25,28, 3, 6, 9,12,15,18, 21,24,27)$
$ 4, 4, 4, 4, 4, 4, 4, 1 $ $29$ $4$ $( 2, 9,16,23)( 3,10,17,24)( 4,11,18,25)( 5,12,19,26)( 6,13,20,27)( 7,14,21,28) ( 8,15,22,29)$
$ 28, 1 $ $29$ $28$ $( 2,17, 4,19, 6,21, 8,23,10,25,12,27,14,29,16, 3,18, 5,20, 7,22, 9,24,11,26, 13,28,15)$
$ 28, 1 $ $29$ $28$ $( 2,13,24, 7,18,29,12,23, 6,17,28,11,22, 5,16,27,10,21, 4,15,26, 9,20, 3,14, 25, 8,19)$
$ 28, 1 $ $29$ $28$ $( 2,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3)$
$ 28, 1 $ $29$ $28$ $( 2,21,12, 3,22,13, 4,23,14, 5,24,15, 6,25,16, 7,26,17, 8,27,18, 9,28,19,10, 29,20,11)$
$ 28, 1 $ $29$ $28$ $( 2,25,20,15,10, 5,28,23,18,13, 8, 3,26,21,16,11, 6,29,24,19,14, 9, 4,27,22, 17,12, 7)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $812=2^{2} \cdot 7 \cdot 29$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  812.7
magma: IdentifyGroup(G);
 
Character table:

1A 2A 4A1 4A-1 7A1 7A-1 7A2 7A-2 7A3 7A-3 14A1 14A-1 14A3 14A-3 14A5 14A-5 28A1 28A-1 28A3 28A-3 28A5 28A-5 28A9 28A-9 28A11 28A-11 28A13 28A-13 29A
Size 1 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 28
2 P 1A 1A 2A 2A 7A2 7A1 7A-3 7A3 7A-2 7A-1 7A3 7A-1 7A-2 7A-3 7A1 7A2 14A3 14A-1 14A1 14A-3 14A3 14A-3 14A-5 14A5 14A-1 14A5 14A-5 14A1 29A
7 P 1A 2A 4A-1 4A1 7A3 7A-2 7A-1 7A1 7A-3 7A2 14A-5 14A-3 14A1 14A5 14A3 14A-1 28A9 28A11 28A3 28A5 28A-5 28A-9 28A-1 28A1 28A-3 28A-13 28A13 28A-11 29A
29 P 1A 2A 4A-1 4A1 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 4A-1 4A1 4A1 4A-1 4A1 4A1 4A1 4A-1 4A-1 4A1 4A-1 4A-1 29A
Type
812.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
812.7.1b R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 1
812.7.1c1 C 1 −1 −i i 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 i −i −i i i −i i −i −i i i −i 1
812.7.1c2 C 1 −1 i −i 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −i i i −i −i i −i i i −i −i i 1
812.7.1d1 C 1 1 1 1 ζ7−3 ζ73 ζ7 ζ7−1 ζ7−2 ζ72 ζ72 ζ7−2 ζ7−1 ζ7 ζ73 ζ7−3 ζ7 ζ7−1 ζ73 ζ7−3 ζ7−2 ζ72 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−1 ζ7 1
812.7.1d2 C 1 1 1 1 ζ73 ζ7−3 ζ7−1 ζ7 ζ72 ζ7−2 ζ7−2 ζ72 ζ7 ζ7−1 ζ7−3 ζ73 ζ7−1 ζ7 ζ7−3 ζ73 ζ72 ζ7−2 ζ7−2 ζ72 ζ73 ζ7−3 ζ7 ζ7−1 1
812.7.1d3 C 1 1 1 1 ζ7−2 ζ72 ζ73 ζ7−3 ζ7 ζ7−1 ζ7−1 ζ7 ζ7−3 ζ73 ζ72 ζ7−2 ζ73 ζ7−3 ζ72 ζ7−2 ζ7 ζ7−1 ζ7−1 ζ7 ζ7−2 ζ72 ζ7−3 ζ73 1
812.7.1d4 C 1 1 1 1 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−1 ζ7 ζ7 ζ7−1 ζ73 ζ7−3 ζ7−2 ζ72 ζ7−3 ζ73 ζ7−2 ζ72 ζ7−1 ζ7 ζ7 ζ7−1 ζ72 ζ7−2 ζ73 ζ7−3 1
812.7.1d5 C 1 1 1 1 ζ7−1 ζ7 ζ7−2 ζ72 ζ7−3 ζ73 ζ73 ζ7−3 ζ72 ζ7−2 ζ7 ζ7−1 ζ7−2 ζ72 ζ7 ζ7−1 ζ7−3 ζ73 ζ73 ζ7−3 ζ7−1 ζ7 ζ72 ζ7−2 1
812.7.1d6 C 1 1 1 1 ζ7 ζ7−1 ζ72 ζ7−2 ζ73 ζ7−3 ζ7−3 ζ73 ζ7−2 ζ72 ζ7−1 ζ7 ζ72 ζ7−2 ζ7−1 ζ7 ζ73 ζ7−3 ζ7−3 ζ73 ζ7 ζ7−1 ζ7−2 ζ72 1
812.7.1e1 C 1 1 −1 −1 ζ7−3 ζ73 ζ7 ζ7−1 ζ7−2 ζ72 ζ72 ζ7−2 ζ7−1 ζ7 ζ73 ζ7−3 −ζ7 −ζ7−1 −ζ73 −ζ7−3 −ζ7−2 −ζ72 −ζ72 −ζ7−2 −ζ7−3 −ζ73 −ζ7−1 −ζ7 1
812.7.1e2 C 1 1 −1 −1 ζ73 ζ7−3 ζ7−1 ζ7 ζ72 ζ7−2 ζ7−2 ζ72 ζ7 ζ7−1 ζ7−3 ζ73 −ζ7−1 −ζ7 −ζ7−3 −ζ73 −ζ72 −ζ7−2 −ζ7−2 −ζ72 −ζ73 −ζ7−3 −ζ7 −ζ7−1 1
812.7.1e3 C 1 1 −1 −1 ζ7−2 ζ72 ζ73 ζ7−3 ζ7 ζ7−1 ζ7−1 ζ7 ζ7−3 ζ73 ζ72 ζ7−2 −ζ73 −ζ7−3 −ζ72 −ζ7−2 −ζ7 −ζ7−1 −ζ7−1 −ζ7 −ζ7−2 −ζ72 −ζ7−3 −ζ73 1
812.7.1e4 C 1 1 −1 −1 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−1 ζ7 ζ7 ζ7−1 ζ73 ζ7−3 ζ7−2 ζ72 −ζ7−3 −ζ73 −ζ7−2 −ζ72 −ζ7−1 −ζ7 −ζ7 −ζ7−1 −ζ72 −ζ7−2 −ζ73 −ζ7−3 1
812.7.1e5 C 1 1 −1 −1 ζ7−1 ζ7 ζ7−2 ζ72 ζ7−3 ζ73 ζ73 ζ7−3 ζ72 ζ7−2 ζ7 ζ7−1 −ζ7−2 −ζ72 −ζ7 −ζ7−1 −ζ7−3 −ζ73 −ζ73 −ζ7−3 −ζ7−1 −ζ7 −ζ72 −ζ7−2 1
812.7.1e6 C 1 1 −1 −1 ζ7 ζ7−1 ζ72 ζ7−2 ζ73 ζ7−3 ζ7−3 ζ73 ζ7−2 ζ72 ζ7−1 ζ7 −ζ72 −ζ7−2 −ζ7−1 −ζ7 −ζ73 −ζ7−3 −ζ7−3 −ζ73 −ζ7 −ζ7−1 −ζ7−2 −ζ72 1
812.7.1f1 C 1 −1 −ζ287 ζ287 −ζ282 ζ2812 ζ284 −ζ2810 −ζ286 ζ288 −ζ288 ζ286 ζ2810 −ζ284 −ζ2812 ζ282 ζ2811 −ζ283 ζ285 −ζ289 −ζ2813 ζ28 −ζ28 ζ2813 ζ289 −ζ285 ζ283 −ζ2811 1
812.7.1f2 C 1 −1 ζ287 −ζ287 ζ2812 −ζ282 −ζ2810 ζ284 ζ288 −ζ286 ζ286 −ζ288 −ζ284 ζ2810 ζ282 −ζ2812 −ζ283 ζ2811 −ζ289 ζ285 ζ28 −ζ2813 ζ2813 −ζ28 −ζ285 ζ289 −ζ2811 ζ283 1
812.7.1f3 C 1 −1 −ζ287 ζ287 ζ2812 −ζ282 −ζ2810 ζ284 ζ288 −ζ286 ζ286 −ζ288 −ζ284 ζ2810 ζ282 −ζ2812 ζ283 −ζ2811 ζ289 −ζ285 −ζ28 ζ2813 −ζ2813 ζ28 ζ285 −ζ289 ζ2811 −ζ283 1
812.7.1f4 C 1 −1 ζ287 −ζ287 −ζ282 ζ2812 ζ284 −ζ2810 −ζ286 ζ288 −ζ288 ζ286 ζ2810 −ζ284 −ζ2812 ζ282 −ζ2811 ζ283 −ζ285 ζ289 ζ2813 −ζ28 ζ28 −ζ2813 −ζ289 ζ285 −ζ283 ζ2811 1
812.7.1f5 C 1 −1 −ζ287 ζ287 −ζ286 ζ288 ζ2812 −ζ282 ζ284 −ζ2810 ζ2810 −ζ284 ζ282 −ζ2812 −ζ288 ζ286 −ζ285 ζ289 ζ28 −ζ2813 ζ2811 −ζ283 ζ283 −ζ2811 ζ2813 −ζ28 −ζ289 ζ285 1
812.7.1f6 C 1 −1 ζ287 −ζ287 ζ288 −ζ286 −ζ282 ζ2812 −ζ2810 ζ284 −ζ284 ζ2810 −ζ2812 ζ282 ζ286 −ζ288 ζ289 −ζ285 −ζ2813 ζ28 −ζ283 ζ2811 −ζ2811 ζ283 −ζ28 ζ2813 ζ285 −ζ289 1
812.7.1f7 C 1 −1 −ζ287 ζ287 ζ288 −ζ286 −ζ282 ζ2812 −ζ2810 ζ284 −ζ284 ζ2810 −ζ2812 ζ282 ζ286 −ζ288 −ζ289 ζ285 ζ2813 −ζ28 ζ283 −ζ2811 ζ2811 −ζ283 ζ28 −ζ2813 −ζ285 ζ289 1
812.7.1f8 C 1 −1 ζ287 −ζ287 −ζ286 ζ288 ζ2812 −ζ282 ζ284 −ζ2810 ζ2810 −ζ284 ζ282 −ζ2812 −ζ288 ζ286 ζ285 −ζ289 −ζ28 ζ2813 −ζ2811 ζ283 −ζ283 ζ2811 −ζ2813 ζ28 ζ289 −ζ285 1
812.7.1f9 C 1 −1 −ζ287 ζ287 −ζ2810 ζ284 −ζ286 ζ288 −ζ282 ζ2812 −ζ2812 ζ282 −ζ288 ζ286 −ζ284 ζ2810 −ζ2813 ζ28 −ζ2811 ζ283 −ζ289 ζ285 −ζ285 ζ289 −ζ283 ζ2811 −ζ28 ζ2813 1
812.7.1f10 C 1 −1 ζ287 −ζ287 ζ284 −ζ2810 ζ288 −ζ286 ζ2812 −ζ282 ζ282 −ζ2812 ζ286 −ζ288 ζ2810 −ζ284 ζ28 −ζ2813 ζ283 −ζ2811 ζ285 −ζ289 ζ289 −ζ285 ζ2811 −ζ283 ζ2813 −ζ28 1
812.7.1f11 C 1 −1 −ζ287 ζ287 ζ284 −ζ2810 ζ288 −ζ286 ζ2812 −ζ282 ζ282 −ζ2812 ζ286 −ζ288 ζ2810 −ζ284 −ζ28 ζ2813 −ζ283 ζ2811 −ζ285 ζ289 −ζ289 ζ285 −ζ2811 ζ283 −ζ2813 ζ28 1
812.7.1f12 C 1 −1 ζ287 −ζ287 −ζ2810 ζ284 −ζ286 ζ288 −ζ282 ζ2812 −ζ2812 ζ282 −ζ288 ζ286 −ζ284 ζ2810 ζ2813 −ζ28 ζ2811 −ζ283 ζ289 −ζ285 ζ285 −ζ289 ζ283 −ζ2811 ζ28 −ζ2813 1
812.7.28a R 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1

magma: CharacterTable(G);