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Magma
magma: G := TransitiveGroup(35, 11);
Group action invariants
Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $11$ | 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: | (1,7,21,32,26,12)(2,6,22,31,27,11)(3,10,23,35,28,15)(4,9,24,34,29,14)(5,8,25,33,30,13)(16,17)(18,20), (1,16)(2,20)(3,19)(4,18)(5,17)(6,11)(7,15)(8,14)(9,13)(10,12)(21,31)(22,35)(23,34)(24,33)(25,32)(27,30)(28,29) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $10$: $D_{5}$ $30$: $D_5\times C_3$ $42$: $F_7$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $D_{5}$
Degree 7: $F_7$
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 $ | $7$ | $3$ | $( 6,21,11)( 7,22,12)( 8,23,13)( 9,24,14)(10,25,15)(16,26,31)(17,27,32) (18,28,33)(19,29,34)(20,30,35)$ |
$ 6, 6, 6, 6, 6, 2, 2, 1 $ | $35$ | $6$ | $( 2, 5)( 3, 4)( 6,16,11,31,21,26)( 7,20,12,35,22,30)( 8,19,13,34,23,29) ( 9,18,14,33,24,28)(10,17,15,32,25,27)$ |
$ 6, 6, 6, 6, 6, 2, 2, 1 $ | $35$ | $6$ | $( 2, 5)( 3, 4)( 6,26,21,31,11,16)( 7,30,22,35,12,20)( 8,29,23,34,13,19) ( 9,28,24,33,14,18)(10,27,25,32,15,17)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $35$ | $2$ | $( 2, 5)( 3, 4)( 6,31)( 7,35)( 8,34)( 9,33)(10,32)(11,26)(12,30)(13,29)(14,28) (15,27)(16,21)(17,25)(18,24)(19,23)(20,22)$ |
$ 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 $ | $6$ | $7$ | $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33) ( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$ |
$ 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 $ | $6$ | $35$ | $( 1, 8,15,17,24,26,33, 5, 7,14,16,23,30,32, 4, 6,13,20,22,29,31, 3,10,12,19, 21,28,35, 2, 9,11,18,25,27,34)$ |
$ 35 $ | $6$ | $35$ | $( 1, 9,12,20,23,26,34, 2,10,13,16,24,27,35, 3, 6,14,17,25,28,31, 4, 7,15,18, 21,29,32, 5, 8,11,19,22,30,33)$ |
$ 35 $ | $6$ | $35$ | $( 1,10,14,18,22,26,35, 4, 8,12,16,25,29,33, 2, 6,15,19,23,27,31, 5, 9,13,17, 21,30,34, 3, 7,11,20,24,28,32)$ |
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: | 210.3 | magma: IdentifyGroup(G);
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Character table: |
2 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 1a 3a 3b 6a 6b 2a 5a 15a 15b 5b 15c 15d 7a 35a 35b 35c 35d 2P 1a 3b 3a 3a 3b 1a 5b 15d 15c 5a 15b 15a 7a 35b 35d 35a 35c 3P 1a 1a 1a 2a 2a 2a 5b 5b 5b 5a 5a 5a 7a 35b 35d 35a 35c 5P 1a 3b 3a 6b 6a 2a 1a 3b 3a 1a 3b 3a 7a 7a 7a 7a 7a 7P 1a 3a 3b 6a 6b 2a 5b 15c 15d 5a 15a 15b 1a 5b 5a 5a 5b 11P 1a 3b 3a 6b 6a 2a 5a 15b 15a 5b 15d 15c 7a 35a 35b 35c 35d 13P 1a 3a 3b 6a 6b 2a 5b 15c 15d 5a 15a 15b 7a 35b 35d 35a 35c 17P 1a 3b 3a 6b 6a 2a 5b 15d 15c 5a 15b 15a 7a 35c 35a 35d 35b 19P 1a 3a 3b 6a 6b 2a 5a 15a 15b 5b 15c 15d 7a 35a 35b 35c 35d 23P 1a 3b 3a 6b 6a 2a 5b 15d 15c 5a 15b 15a 7a 35c 35a 35d 35b 29P 1a 3b 3a 6b 6a 2a 5a 15b 15a 5b 15d 15c 7a 35d 35c 35b 35a 31P 1a 3a 3b 6a 6b 2a 5a 15a 15b 5b 15c 15d 7a 35d 35c 35b 35a X.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 X.3 1 A /A -/A -A -1 1 A /A 1 A /A 1 1 1 1 1 X.4 1 /A A -A -/A -1 1 /A A 1 /A A 1 1 1 1 1 X.5 1 A /A /A A 1 1 A /A 1 A /A 1 1 1 1 1 X.6 1 /A A A /A 1 1 /A A 1 /A A 1 1 1 1 1 X.7 2 B /B . . . C E /E *C F /F 2 C *C *C C X.8 2 /B B . . . C /E E *C /F F 2 C *C *C C X.9 2 B /B . . . *C F /F C E /E 2 *C C C *C X.10 2 /B B . . . *C /F F C /E E 2 *C C C *C X.11 2 2 2 . . . C C C *C *C *C 2 C *C *C C X.12 2 2 2 . . . *C *C *C C C C 2 *C C C *C X.13 6 . . . . . 6 . . 6 . . -1 -1 -1 -1 -1 X.14 6 . . . . . D . . *D . . -1 G I J H X.15 6 . . . . . D . . *D . . -1 H J I G X.16 6 . . . . . *D . . D . . -1 I H G J X.17 6 . . . . . *D . . D . . -1 J G H 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)^2+3*E(5)^3 = (-3-3*Sqrt(5))/2 = -3-3b5 E = E(15)^2+E(15)^8 F = E(15)^11+E(15)^14 G = E(35)^4+E(35)^6+E(35)^9+E(35)^26+E(35)^29+E(35)^31 H = E(35)+E(35)^11+E(35)^16+E(35)^19+E(35)^24+E(35)^34 I = E(35)^8+E(35)^12+E(35)^17+E(35)^18+E(35)^23+E(35)^27 J = E(35)^2+E(35)^3+E(35)^13+E(35)^22+E(35)^32+E(35)^33 |
magma: CharacterTable(G);