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Magma
magma: G := TransitiveGroup(42, 39);
Group action invariants
Degree $n$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $A_4\times C_7:C_3$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,14,25,39,9,23,36,3,16,28,41,12,20,32,5,18,30,38,7,21,33)(2,13,26,40,10,24,35,4,15,27,42,11,19,31,6,17,29,37,8,22,34), (1,21,15)(2,22,16)(3,24,18)(4,23,17)(5,19,13)(6,20,14)(7,32,37)(8,31,38)(9,34,39)(10,33,40)(11,36,41)(12,35,42)(25,27,29)(26,28,30) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ x 4 $9$: $C_3^2$ $12$: $A_4$ $21$: $C_7:C_3$ $36$: $C_3\times A_4$ $63$: 21T7 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 6: $A_4$
Degree 7: $C_7:C_3$
Degree 14: None
Degree 21: 21T7
Low degree siblings
28T40Siblings are shown 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, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 7,18,28)( 8,17,27)( 9,14,30)(10,13,29)(11,15,26)(12,16,25)(19,40,35) (20,39,36)(21,41,32)(22,42,31)(23,38,33)(24,37,34)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 7,28,18)( 8,27,17)( 9,30,14)(10,29,13)(11,26,15)(12,25,16)(19,35,40) (20,36,39)(21,32,41)(22,31,42)(23,33,38)(24,34,37)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 3, 4)( 5, 6)( 9,10)(11,12)(13,14)(15,16)(21,22)(23,24)(25,26)(29,30)(31,32) (33,34)(37,38)(41,42)$ |
$ 6, 6, 6, 6, 3, 3, 3, 3, 2, 2, 1, 1 $ | $21$ | $6$ | $( 3, 4)( 5, 6)( 7,18,28)( 8,17,27)( 9,13,30,10,14,29)(11,16,26,12,15,25) (19,40,35)(20,39,36)(21,42,32,22,41,31)(23,37,33,24,38,34)$ |
$ 6, 6, 6, 6, 3, 3, 3, 3, 2, 2, 1, 1 $ | $21$ | $6$ | $( 3, 4)( 5, 6)( 7,28,18)( 8,27,17)( 9,29,14,10,30,13)(11,25,15,12,26,16) (19,35,40)(20,36,39)(21,31,41,22,32,42)(23,34,38,24,33,37)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7, 9,12)( 8,10,11)(13,15,17)(14,16,18)(19,22,24) (20,21,23)(25,28,30)(26,27,29)(31,34,35)(32,33,36)(37,40,42)(38,39,41)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $28$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,14,25)( 8,13,26)( 9,16,28)(10,15,27)(11,17,29) (12,18,30)(19,42,34)(20,41,33)(21,38,36)(22,37,35)(23,39,32)(24,40,31)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $28$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,30,16)( 8,29,15)( 9,25,18)(10,26,17)(11,27,13) (12,28,14)(19,31,37)(20,32,38)(21,33,39)(22,34,40)(23,36,41)(24,35,42)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,12, 9)( 8,11,10)(13,17,15)(14,18,16)(19,24,22) (20,23,21)(25,30,28)(26,29,27)(31,35,34)(32,36,33)(37,42,40)(38,41,39)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $28$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,16,30)( 8,15,29)( 9,18,25)(10,17,26)(11,13,27) (12,14,28)(19,37,31)(20,38,32)(21,39,33)(22,40,34)(23,41,36)(24,42,35)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $28$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,25,14)( 8,26,13)( 9,28,16)(10,27,15)(11,29,17) (12,30,18)(19,34,42)(20,33,41)(21,36,38)(22,35,37)(23,32,39)(24,31,40)$ |
$ 7, 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1, 7,18,20,28,36,39)( 2, 8,17,19,27,35,40)( 3, 9,14,21,30,32,41) ( 4,10,13,22,29,31,42)( 5,12,16,23,25,33,38)( 6,11,15,24,26,34,37)$ |
$ 14, 14, 7, 7 $ | $9$ | $14$ | $( 1, 7,18,20,28,36,39)( 2, 8,17,19,27,35,40)( 3,10,14,22,30,31,41, 4, 9,13,21, 29,32,42)( 5,11,16,24,25,34,38, 6,12,15,23,26,33,37)$ |
$ 21, 21 $ | $12$ | $21$ | $( 1, 9,15,20,30,34,39, 3,11,18,21,26,36,41, 6, 7,14,24,28,32,37) ( 2,10,16,19,29,33,40, 4,12,17,22,25,35,42, 5, 8,13,23,27,31,38)$ |
$ 21, 21 $ | $12$ | $21$ | $( 1,11,14,20,26,32,39, 6, 9,18,24,30,36,37, 3, 7,15,21,28,34,41) ( 2,12,13,19,25,31,40, 5,10,17,23,29,35,38, 4, 8,16,22,27,33,42)$ |
$ 14, 14, 7, 7 $ | $9$ | $14$ | $( 1,19,39,17,36, 8,28, 2,20,40,18,35, 7,27)( 3,21,41,14,32, 9,30) ( 4,22,42,13,31,10,29)( 5,24,38,15,33,11,25, 6,23,37,16,34,12,26)$ |
$ 7, 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1,20,39,18,36, 7,28)( 2,19,40,17,35, 8,27)( 3,21,41,14,32, 9,30) ( 4,22,42,13,31,10,29)( 5,23,38,16,33,12,25)( 6,24,37,15,34,11,26)$ |
$ 21, 21 $ | $12$ | $21$ | $( 1,21,38,18,32,12,28, 3,23,39,14,33, 7,30, 5,20,41,16,36, 9,25) ( 2,22,37,17,31,11,27, 4,24,40,13,34, 8,29, 6,19,42,15,35,10,26)$ |
$ 21, 21 $ | $12$ | $21$ | $( 1,23,42,18,33,10,28, 5,22,39,16,31, 7,25, 4,20,38,13,36,12,29) ( 2,24,41,17,34, 9,27, 6,21,40,15,32, 8,26, 3,19,37,14,35,11,30)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $252=2^{2} \cdot 3^{2} \cdot 7$ | 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: | 252.27 | magma: IdentifyGroup(G);
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Character table: |
2 2 2 2 2 2 2 . . . . . . 2 2 . . 2 2 . . 3 2 2 2 1 1 1 2 2 2 2 2 2 1 . 1 1 . 1 1 1 7 1 . . 1 . . 1 . . 1 . . 1 1 1 1 1 1 1 1 1a 3a 3b 2a 6a 6b 3c 3d 3e 3f 3g 3h 7a 14a 21a 21b 14b 7b 21c 21d 2P 1a 3b 3a 1a 3b 3a 3f 3h 3g 3c 3e 3d 7a 7a 21b 21a 7b 7b 21d 21c 3P 1a 1a 1a 2a 2a 2a 1a 1a 1a 1a 1a 1a 7b 14b 7b 7b 14a 7a 7a 7a 5P 1a 3b 3a 2a 6b 6a 3f 3h 3g 3c 3e 3d 7b 14b 21d 21c 14a 7a 21b 21a 7P 1a 3a 3b 2a 6a 6b 3c 3d 3e 3f 3g 3h 1a 2a 3c 3f 2a 1a 3c 3f 11P 1a 3b 3a 2a 6b 6a 3f 3h 3g 3c 3e 3d 7a 14a 21b 21a 14b 7b 21d 21c 13P 1a 3a 3b 2a 6a 6b 3c 3d 3e 3f 3g 3h 7b 14b 21c 21d 14a 7a 21a 21b 17P 1a 3b 3a 2a 6b 6a 3f 3h 3g 3c 3e 3d 7b 14b 21d 21c 14a 7a 21b 21a 19P 1a 3a 3b 2a 6a 6b 3c 3d 3e 3f 3g 3h 7b 14b 21c 21d 14a 7a 21a 21b X.1 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 A A A /A /A /A 1 1 A /A 1 1 A /A X.3 1 1 1 1 1 1 /A /A /A A A A 1 1 /A A 1 1 /A A X.4 1 A /A 1 A /A 1 A /A 1 A /A 1 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 1 X.6 1 A /A 1 A /A A /A 1 /A 1 A 1 1 A /A 1 1 A /A X.7 1 /A A 1 /A A /A A 1 A 1 /A 1 1 /A A 1 1 /A A X.8 1 A /A 1 A /A /A 1 A A /A 1 1 1 /A A 1 1 /A A X.9 1 /A A 1 /A A A 1 /A /A A 1 1 1 A /A 1 1 A /A X.10 3 3 3 -1 -1 -1 . . . . . . 3 -1 . . -1 3 . . X.11 3 . . 3 . . 3 . . 3 . . C C C C /C /C /C /C X.12 3 . . 3 . . 3 . . 3 . . /C /C /C /C C C C C X.13 3 . . 3 . . B . . /B . . C C E F /C /C /F /E X.14 3 . . 3 . . /B . . B . . C C F E /C /C /E /F X.15 3 . . 3 . . B . . /B . . /C /C /F /E C C E F X.16 3 . . 3 . . /B . . B . . /C /C /E /F C C F E X.17 3 B /B -1 -A -/A . . . . . . 3 -1 . . -1 3 . . X.18 3 /B B -1 -/A -A . . . . . . 3 -1 . . -1 3 . . X.19 9 . . -3 . . . . . . . . D -C . . -/C /D . . X.20 9 . . -3 . . . . . . . . /D -/C . . -C D . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 3*E(3)^2 = (-3-3*Sqrt(-3))/2 = -3-3b3 C = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7 D = 3*E(7)^3+3*E(7)^5+3*E(7)^6 = (-3-3*Sqrt(-7))/2 = -3-3b7 E = E(21)^2+E(21)^8+E(21)^11 F = E(21)+E(21)^4+E(21)^16 |
magma: CharacterTable(G);