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Magma
magma: G := TransitiveGroup(35, 16);
Group action invariants
Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{35}:C_{12}$ | ||
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: | (1,33,30,22,19,11,8,5,32,29,21,18,15,7,4,31,28,25,17,14,6,3,35,27,24,16,13,10,2,34,26,23,20,12,9), (1,16,21,11,31,26)(2,18,25,14,32,28,5,19,22,13,35,29)(3,20,24,12,33,30,4,17,23,15,34,27)(7,8,10,9) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $4$: $C_4$ $6$: $C_6$ $12$: $C_{12}$ $20$: $F_5$ $42$: $F_7$ $60$: $F_5\times C_3$ $84$: 28T12 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $F_5$
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 $ | $7$ | $3$ | $( 6,21,11)( 7,22,12)( 8,23,13)( 9,24,14)(10,25,15)(16,26,31)(17,27,32) (18,28,33)(19,29,34)(20,30,35)$ |
$ 12, 12, 6, 4, 1 $ | $35$ | $12$ | $( 2, 3, 5, 4)( 6,16,11,31,21,26)( 7,18,15,34,22,28,10,19,12,33,25,29) ( 8,20,14,32,23,30, 9,17,13,35,24,27)$ |
$ 12, 12, 6, 4, 1 $ | $35$ | $12$ | $( 2, 3, 5, 4)( 6,26,21,31,11,16)( 7,28,25,34,12,18,10,29,22,33,15,19) ( 8,30,24,32,13,20, 9,27,23,35,14,17)$ |
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $35$ | $4$ | $( 2, 3, 5, 4)( 6,31)( 7,33,10,34)( 8,35, 9,32)(11,26)(12,28,15,29) (13,30,14,27)(16,21)(17,23,20,24)(18,25,19,22)$ |
$ 12, 12, 6, 4, 1 $ | $35$ | $12$ | $( 2, 4, 5, 3)( 6,16,11,31,21,26)( 7,19,15,33,22,29,10,18,12,34,25,28) ( 8,17,14,35,23,27, 9,20,13,32,24,30)$ |
$ 12, 12, 6, 4, 1 $ | $35$ | $12$ | $( 2, 4, 5, 3)( 6,26,21,31,11,16)( 7,29,25,33,12,19,10,28,22,34,15,18) ( 8,27,24,35,13,17, 9,30,23,32,14,20)$ |
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ | $35$ | $4$ | $( 2, 4, 5, 3)( 6,31)( 7,34,10,33)( 8,32, 9,35)(11,26)(12,29,15,28) (13,27,14,30)(16,21)(17,24,20,23)(18,22,19,25)$ |
$ 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 $ | $4$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$ |
$ 15, 15, 5 $ | $28$ | $15$ | $( 1, 2, 3, 4, 5)( 6,12,23, 9,15,21, 7,13,24,10,11,22, 8,14,25)(16,32,28,19,35, 26,17,33,29,20,31,27,18,34,30)$ |
$ 15, 15, 5 $ | $28$ | $15$ | $( 1, 2, 3, 4, 5)( 6,22,13, 9,25,11, 7,23,14,10,21,12, 8,24,15)(16,27,33,19,30, 31,17,28,34,20,26,32,18,29,35)$ |
$ 7, 7, 7, 7, 7 $ | $6$ | $7$ | $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33) ( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$ |
$ 14, 14, 7 $ | $30$ | $14$ | $( 1, 6,11,16,21,26,31)( 2,10,12,20,22,30,32, 5, 7,15,17,25,27,35) ( 3, 9,13,19,23,29,33, 4, 8,14,18,24,28,34)$ |
$ 35 $ | $12$ | $35$ | $( 1, 7,13,19,25,26,32, 3, 9,15,16,22,28,34, 5, 6,12,18,24,30,31, 2, 8,14,20, 21,27,33, 4,10,11,17,23,29,35)$ |
$ 35 $ | $12$ | $35$ | $( 1, 8,15,17,24,26,33, 5, 7,14,16,23,30,32, 4, 6,13,20,22,29,31, 3,10,12,19, 21,28,35, 2, 9,11,18,25,27,34)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $420=2^{2} \cdot 3 \cdot 5 \cdot 7$ | 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: | 420.15 | magma: IdentifyGroup(G);
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Character table: |
2 2 2 2 2 2 2 2 2 2 2 2 2 . . . 1 1 . . 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . . . . 5 1 1 1 . . . . . . . . . 1 1 1 1 . 1 1 7 1 . . . . . . . . 1 . . 1 . . 1 1 1 1 1a 3a 3b 12a 12b 4a 12c 12d 4b 2a 6a 6b 5a 15a 15b 7a 14a 35a 35b 2P 1a 3b 3a 6a 6b 2a 6a 6b 2a 1a 3b 3a 5a 15b 15a 7a 7a 35b 35a 3P 1a 1a 1a 4b 4b 4b 4a 4a 4a 2a 2a 2a 5a 5a 5a 7a 14a 35a 35b 5P 1a 3b 3a 12b 12a 4a 12d 12c 4b 2a 6b 6a 1a 3b 3a 7a 14a 7a 7a 7P 1a 3a 3b 12c 12d 4b 12a 12b 4a 2a 6a 6b 5a 15a 15b 1a 2a 5a 5a 11P 1a 3b 3a 12d 12c 4b 12b 12a 4a 2a 6b 6a 5a 15b 15a 7a 14a 35a 35b 13P 1a 3a 3b 12a 12b 4a 12c 12d 4b 2a 6a 6b 5a 15a 15b 7a 14a 35a 35b 17P 1a 3b 3a 12b 12a 4a 12d 12c 4b 2a 6b 6a 5a 15b 15a 7a 14a 35a 35b 19P 1a 3a 3b 12c 12d 4b 12a 12b 4a 2a 6a 6b 5a 15a 15b 7a 14a 35b 35a 23P 1a 3b 3a 12d 12c 4b 12b 12a 4a 2a 6b 6a 5a 15b 15a 7a 14a 35b 35a 29P 1a 3b 3a 12b 12a 4a 12d 12c 4b 2a 6b 6a 5a 15b 15a 7a 14a 35a 35b 31P 1a 3a 3b 12c 12d 4b 12a 12b 4a 2a 6a 6b 5a 15a 15b 7a 14a 35b 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 1 1 C C C -C -C -C -1 -1 -1 1 1 1 1 -1 1 1 X.4 1 1 1 -C -C -C C C C -1 -1 -1 1 1 1 1 -1 1 1 X.5 1 A /A -/A -A -1 -/A -A -1 1 A /A 1 A /A 1 1 1 1 X.6 1 /A A -A -/A -1 -A -/A -1 1 /A A 1 /A A 1 1 1 1 X.7 1 A /A /A A 1 /A A 1 1 A /A 1 A /A 1 1 1 1 X.8 1 /A A A /A 1 A /A 1 1 /A A 1 /A A 1 1 1 1 X.9 1 A /A D -/D C -D /D -C -1 -A -/A 1 A /A 1 -1 1 1 X.10 1 A /A -D /D -C D -/D C -1 -A -/A 1 A /A 1 -1 1 1 X.11 1 /A A -/D D C /D -D -C -1 -/A -A 1 /A A 1 -1 1 1 X.12 1 /A A /D -D -C -/D D C -1 -/A -A 1 /A A 1 -1 1 1 X.13 4 4 4 . . . . . . . . . -1 -1 -1 4 . -1 -1 X.14 4 B /B . . . . . . . . . -1 -A -/A 4 . -1 -1 X.15 4 /B B . . . . . . . . . -1 -/A -A 4 . -1 -1 X.16 6 . . . . . . . . -6 . . 6 . . -1 1 -1 -1 X.17 6 . . . . . . . . 6 . . 6 . . -1 -1 -1 -1 X.18 12 . . . . . . . . . . . -3 . . -2 . E /E X.19 12 . . . . . . . . . . . -3 . . -2 . /E E A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 4*E(3)^2 = -2-2*Sqrt(-3) = -2-2i3 C = -E(4) = -Sqrt(-1) = -i D = -E(12)^7 E = E(35)^2+E(35)^6+E(35)^8+E(35)^18+E(35)^19+E(35)^22+E(35)^23+E(35)^24+E(35)^26+E(35)^31+E(35)^32+E(35)^34 = (1-Sqrt(-35))/2 = -b35 |
magma: CharacterTable(G);