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Magma
magma: G := TransitiveGroup(35, 9);
Group action invariants
Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $9$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{35}:C_6$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,35,14,3,32,11,5,34,13,2,31,15,4,33,12)(6,20,24,8,17,21,10,19,23,7,16,25,9,18,22)(26,30,29,28,27), (1,12,21,32,6,17,26,2,11,22,31,7,16,27)(3,15,23,35,8,20,28,5,13,25,33,10,18,30)(4,14,24,34,9,19,29) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $10$: $D_{5}$ $21$: $C_7:C_3$ $30$: $D_5\times C_3$ $42$: $(C_7:C_3) \times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $D_{5}$
Degree 7: $C_7:C_3$
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
Cycle Type | Size | Order | Representative |
$ 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, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 6,11,21)( 7,12,22)( 8,13,23)( 9,14,24)(10,15,25)(16,31,26)(17,32,27) (18,33,28)(19,34,29)(20,35,30)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 6,21,11)( 7,22,12)( 8,23,13)( 9,24,14)(10,25,15)(16,26,31)(17,27,32) (18,28,33)(19,29,34)(20,30,35)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)$ |
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $35$ | $6$ | $( 2, 5)( 3, 4)( 6,11,21)( 7,15,22,10,12,25)( 8,14,23, 9,13,24)(16,31,26) (17,35,27,20,32,30)(18,34,28,19,33,29)$ |
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ | $35$ | $6$ | $( 2, 5)( 3, 4)( 6,21,11)( 7,25,12,10,22,15)( 8,24,13, 9,23,14)(16,26,31) (17,30,32,20,27,35)(18,29,33,19,28,34)$ |
$ 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$ |
$ 15, 15, 5 $ | $14$ | $15$ | $( 1, 2, 3, 4, 5)( 6,12,23, 9,15,21, 7,13,24,10,11,22, 8,14,25)(16,32,28,19,35, 26,17,33,29,20,31,27,18,34,30)$ |
$ 15, 15, 5 $ | $14$ | $15$ | $( 1, 2, 3, 4, 5)( 6,22,13, 9,25,11, 7,23,14,10,21,12, 8,24,15)(16,27,33,19,30, 31,17,28,34,20,26,32,18,29,35)$ |
$ 5, 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)$ |
$ 15, 15, 5 $ | $14$ | $15$ | $( 1, 3, 5, 2, 4)( 6,13,25, 7,14,21, 8,15,22, 9,11,23,10,12,24)(16,33,30,17,34, 26,18,35,27,19,31,28,20,32,29)$ |
$ 15, 15, 5 $ | $14$ | $15$ | $( 1, 3, 5, 2, 4)( 6,23,15, 7,24,11, 8,25,12, 9,21,13,10,22,14)(16,28,35,17,29, 31,18,30,32,19,26,33,20,27,34)$ |
$ 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33) ( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$ |
$ 14, 14, 7 $ | $15$ | $14$ | $( 1, 6,11,16,21,26,31)( 2,10,12,20,22,30,32, 5, 7,15,17,25,27,35) ( 3, 9,13,19,23,29,33, 4, 8,14,18,24,28,34)$ |
$ 35 $ | $6$ | $35$ | $( 1, 7,13,19,25,26,32, 3, 9,15,16,22,28,34, 5, 6,12,18,24,30,31, 2, 8,14,20, 21,27,33, 4,10,11,17,23,29,35)$ |
$ 35 $ | $6$ | $35$ | $( 1, 8,15,17,24,26,33, 5, 7,14,16,23,30,32, 4, 6,13,20,22,29,31, 3,10,12,19, 21,28,35, 2, 9,11,18,25,27,34)$ |
$ 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1,16,31,11,26, 6,21)( 2,17,32,12,27, 7,22)( 3,18,33,13,28, 8,23) ( 4,19,34,14,29, 9,24)( 5,20,35,15,30,10,25)$ |
$ 14, 14, 7 $ | $15$ | $14$ | $( 1,16,31,11,26, 6,21)( 2,20,32,15,27,10,22, 5,17,35,12,30, 7,25) ( 3,19,33,14,28, 9,23, 4,18,34,13,29, 8,24)$ |
$ 35 $ | $6$ | $35$ | $( 1,17,33,14,30, 6,22, 3,19,35,11,27, 8,24, 5,16,32,13,29,10,21, 2,18,34,15, 26, 7,23, 4,20,31,12,28, 9,25)$ |
$ 35 $ | $6$ | $35$ | $( 1,18,35,12,29, 6,23, 5,17,34,11,28,10,22, 4,16,33,15,27, 9,21, 3,20,32,14, 26, 8,25, 2,19,31,13,30, 7,24)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $210=2 \cdot 3 \cdot 5 \cdot 7$ | magma: Order(G);
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Cyclic: | no | magma: IsCyclic(G);
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Abelian: | no | magma: IsAbelian(G);
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Solvable: | yes | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 210.2 | magma: IdentifyGroup(G);
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Character table: |
2 1 1 1 1 1 1 . . . . . . 1 1 . . 1 1 . 3 1 1 1 1 1 1 1 1 1 1 1 1 . . . . . . . 5 1 1 1 . . . 1 1 1 1 1 1 1 . 1 1 1 . 1 7 1 . . 1 . . 1 . . 1 . . 1 1 1 1 1 1 1 1a 3a 3b 2a 6a 6b 5a 15a 15b 5b 15c 15d 7a 14a 35a 35b 7b 14b 35c 2P 1a 3b 3a 1a 3b 3a 5b 15d 15c 5a 15b 15a 7a 7a 35b 35a 7b 7b 35d 3P 1a 1a 1a 2a 2a 2a 5b 5b 5b 5a 5a 5a 7b 14b 35d 35c 7a 14a 35b 5P 1a 3b 3a 2a 6b 6a 1a 3b 3a 1a 3b 3a 7b 14b 7b 7b 7a 14a 7a 7P 1a 3a 3b 2a 6a 6b 5b 15c 15d 5a 15a 15b 1a 2a 5b 5a 1a 2a 5b 11P 1a 3b 3a 2a 6b 6a 5a 15b 15a 5b 15d 15c 7a 14a 35a 35b 7b 14b 35c 13P 1a 3a 3b 2a 6a 6b 5b 15c 15d 5a 15a 15b 7b 14b 35d 35c 7a 14a 35b 17P 1a 3b 3a 2a 6b 6a 5b 15d 15c 5a 15b 15a 7b 14b 35d 35c 7a 14a 35b 19P 1a 3a 3b 2a 6a 6b 5a 15a 15b 5b 15c 15d 7b 14b 35c 35d 7a 14a 35a 23P 1a 3b 3a 2a 6b 6a 5b 15d 15c 5a 15b 15a 7a 14a 35b 35a 7b 14b 35d 29P 1a 3b 3a 2a 6b 6a 5a 15b 15a 5b 15d 15c 7a 14a 35a 35b 7b 14b 35c 31P 1a 3a 3b 2a 6a 6b 5a 15a 15b 5b 15c 15d 7b 14b 35c 35d 7a 14a 35a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 -1 1 1 1 -1 1 X.3 1 A /A -1 -A -/A 1 A /A 1 A /A 1 -1 1 1 1 -1 1 X.4 1 /A A -1 -/A -A 1 /A A 1 /A A 1 -1 1 1 1 -1 1 X.5 1 A /A 1 A /A 1 A /A 1 A /A 1 1 1 1 1 1 1 X.6 1 /A A 1 /A A 1 /A A 1 /A A 1 1 1 1 1 1 1 X.7 2 B /B . . . C E /E *C F /F 2 . C *C 2 . C X.8 2 /B B . . . C /E E *C /F F 2 . C *C 2 . C X.9 2 B /B . . . *C F /F C E /E 2 . *C C 2 . *C X.10 2 /B B . . . *C /F F C /E E 2 . *C C 2 . *C X.11 2 2 2 . . . C C C *C *C *C 2 . C *C 2 . C X.12 2 2 2 . . . *C *C *C C C C 2 . *C C 2 . *C X.13 3 . . -3 . . 3 . . 3 . . G -G G G /G -/G /G X.14 3 . . -3 . . 3 . . 3 . . /G -/G /G /G G -G G X.15 3 . . 3 . . 3 . . 3 . . G G G G /G /G /G X.16 3 . . 3 . . 3 . . 3 . . /G /G /G /G G G G X.17 6 . . . . . D . . *D . . H . I J /H . /I X.18 6 . . . . . D . . *D . . /H . /I /J H . I X.19 6 . . . . . *D . . D . . H . J I /H . /J X.20 6 . . . . . *D . . D . . /H . /J /I H . J 2 . 3 . 5 1 7 1 35d 2P 35c 3P 35a 5P 7a 7P 5a 11P 35d 13P 35a 17P 35a 19P 35b 23P 35c 29P 35d 31P 35b X.1 1 X.2 1 X.3 1 X.4 1 X.5 1 X.6 1 X.7 *C X.8 *C X.9 C X.10 C X.11 *C X.12 C X.13 /G X.14 G X.15 /G X.16 G X.17 /J X.18 J X.19 /I X.20 I A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3) = -1+Sqrt(-3) = 2b3 C = E(5)+E(5)^4 = (-1+Sqrt(5))/2 = b5 D = 3*E(5)+3*E(5)^4 = (-3+3*Sqrt(5))/2 = 3b5 E = E(15)^2+E(15)^8 F = E(15)^11+E(15)^14 G = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7 H = 2*E(7)^3+2*E(7)^5+2*E(7)^6 = -1-Sqrt(-7) = -1-i7 I = E(35)^2+E(35)^8+E(35)^18+E(35)^22+E(35)^23+E(35)^32 J = E(35)+E(35)^4+E(35)^9+E(35)^11+E(35)^16+E(35)^29 |
magma: CharacterTable(G);