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Magma
magma: G := TransitiveGroup(42, 23);
Group action invariants
Degree $n$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $23$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{21}:C_6$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,9,20)(2,8,21)(3,7,19)(4,35,12)(5,36,10)(6,34,11)(13,31,27)(14,32,26)(15,33,25)(16,17,18)(22,29,40)(23,30,42)(24,28,41), (1,18,15,24,37,36)(2,16,14,22,38,34)(3,17,13,23,39,35)(4,8,41,26,28,19)(5,7,40,27,29,20)(6,9,42,25,30,21)(10,31,12,32,11,33) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ $21$: $C_7:C_3$ $42$: $(C_7:C_3) \times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 6: $S_3\times C_3$
Degree 7: $C_7:C_3$
Degree 14: $(C_7:C_3) \times C_2$
Degree 21: None
Low degree siblings
21T11, 42T19Siblings 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, 3, 1, 1, 1 $ | $14$ | $3$ | $( 4,28,36)( 5,29,34)( 6,30,35)( 7,15,25)( 8,13,27)( 9,14,26)(10,42,16) (11,41,17)(12,40,18)(19,38,33)(20,39,32)(21,37,31)(22,23,24)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $14$ | $3$ | $( 4,36,28)( 5,34,29)( 6,35,30)( 7,25,15)( 8,27,13)( 9,26,14)(10,16,42) (11,17,41)(12,18,40)(19,33,38)(20,32,39)(21,31,37)(22,24,23)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,20,21) (22,24,23)(25,26,27)(28,30,29)(31,33,32)(34,36,35)(37,38,39)(40,41,42)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 2, 3)( 4,35,29)( 5,36,30)( 6,34,28)( 7,26,13)( 8,25,14)( 9,27,15) (10,18,41)(11,16,40)(12,17,42)(19,32,37)(20,31,38)(21,33,39)(22,23,24)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 3, 2)( 4,29,35)( 5,30,36)( 6,28,34)( 7,13,26)( 8,14,25)( 9,15,27) (10,41,18)(11,40,16)(12,42,17)(19,37,32)(20,38,31)(21,39,33)(22,24,23)$ |
$ 14, 14, 14 $ | $9$ | $14$ | $( 1, 4, 9,10,14,17,20,24,26,30,32,34,39,40)( 2, 5, 8,11,13,18,21,22,27,28,31, 35,37,42)( 3, 6, 7,12,15,16,19,23,25,29,33,36,38,41)$ |
$ 6, 6, 6, 6, 6, 6, 6 $ | $21$ | $6$ | $( 1, 4,31,22,25,12)( 2, 5,33,23,26,10)( 3, 6,32,24,27,11)( 7,16,14,30,37,35) ( 8,18,15,29,39,34)( 9,17,13,28,38,36)(19,41,20,40,21,42)$ |
$ 6, 6, 6, 6, 6, 6, 6 $ | $21$ | $6$ | $( 1, 4,38,23,27,18)( 2, 5,39,24,25,16)( 3, 6,37,22,26,17)( 7,29, 8,28, 9,30) (10,19,36,31,42,14)(11,20,34,33,41,13)(12,21,35,32,40,15)$ |
$ 21, 21 $ | $6$ | $21$ | $( 1, 7,13,20,25,31,39, 3, 8,14,19,27,32,38, 2, 9,15,21,26,33,37) ( 4,11,16,24,28,36,40, 5,12,17,22,29,34,42, 6,10,18,23,30,35,41)$ |
$ 7, 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1, 9,14,20,26,32,39)( 2, 8,13,21,27,31,37)( 3, 7,15,19,25,33,38) ( 4,10,17,24,30,34,40)( 5,11,18,22,28,35,42)( 6,12,16,23,29,36,41)$ |
$ 14, 14, 14 $ | $9$ | $14$ | $( 1,10,20,30,39, 4,14,24,32,40, 9,17,26,34)( 2,11,21,28,37, 5,13,22,31,42, 8, 18,27,35)( 3,12,19,29,38, 6,15,23,33,41, 7,16,25,36)$ |
$ 21, 21 $ | $6$ | $21$ | $( 1,19,37,14,33, 8,26, 3,21,39,15,31, 9,25, 2,20,38,13,32, 7,27) ( 4,22,41,17,35,12,30, 5,23,40,18,36,10,28, 6,24,42,16,34,11,29)$ |
$ 7, 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1,20,39,14,32, 9,26)( 2,21,37,13,31, 8,27)( 3,19,38,15,33, 7,25) ( 4,24,40,17,34,10,30)( 5,22,42,18,35,11,28)( 6,23,41,16,36,12,29)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1,22)( 2,23)( 3,24)( 4,25)( 5,26)( 6,27)( 7,30)( 8,29)( 9,28)(10,33)(11,32) (12,31)(13,36)(14,35)(15,34)(16,37)(17,38)(18,39)(19,40)(20,42)(21,41)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $126=2 \cdot 3^{2} \cdot 7$ | 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: | 126.8 | magma: IdentifyGroup(G);
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Character table: |
2 1 . . . 1 1 1 1 1 . 1 1 . 1 1 3 2 2 2 2 2 2 . 1 1 1 1 . 1 1 1 7 1 . . 1 . . 1 . . 1 1 1 1 1 1 1a 3a 3b 3c 3d 3e 14a 6a 6b 21a 7a 14b 21b 7b 2a 2P 1a 3b 3a 3c 3e 3d 7a 3d 3e 21a 7a 7b 21b 7b 1a 3P 1a 1a 1a 1a 1a 1a 14b 2a 2a 7b 7b 14a 7a 7a 2a 5P 1a 3b 3a 3c 3e 3d 14b 6b 6a 21b 7b 14a 21a 7a 2a 7P 1a 3a 3b 3c 3d 3e 2a 6a 6b 3c 1a 2a 3c 1a 2a 11P 1a 3b 3a 3c 3e 3d 14a 6b 6a 21a 7a 14b 21b 7b 2a 13P 1a 3a 3b 3c 3d 3e 14b 6a 6b 21b 7b 14a 21a 7a 2a 17P 1a 3b 3a 3c 3e 3d 14b 6b 6a 21b 7b 14a 21a 7a 2a 19P 1a 3a 3b 3c 3d 3e 14b 6a 6b 21b 7b 14a 21a 7a 2a X.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 X.3 1 A /A 1 /A A -1 -A -/A 1 1 -1 1 1 -1 X.4 1 /A A 1 A /A -1 -/A -A 1 1 -1 1 1 -1 X.5 1 A /A 1 /A A 1 A /A 1 1 1 1 1 1 X.6 1 /A A 1 A /A 1 /A A 1 1 1 1 1 1 X.7 2 -1 -1 -1 2 2 . . . -1 2 . -1 2 . X.8 2 -/A -A -1 B /B . . . -1 2 . -1 2 . X.9 2 -A -/A -1 /B B . . . -1 2 . -1 2 . X.10 3 . . 3 . . C . . -C -C /C -/C -/C -3 X.11 3 . . 3 . . /C . . -/C -/C C -C -C -3 X.12 3 . . 3 . . -/C . . -/C -/C -C -C -C 3 X.13 3 . . 3 . . -C . . -C -C -/C -/C -/C 3 X.14 6 . . -3 . . . . . /C D . C /D . X.15 6 . . -3 . . . . . C /D . /C D . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3 C = -E(7)-E(7)^2-E(7)^4 = (1-Sqrt(-7))/2 = -b7 D = 2*E(7)^3+2*E(7)^5+2*E(7)^6 = -1-Sqrt(-7) = -1-i7 |
magma: CharacterTable(G);