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Magma
magma: G := TransitiveGroup(21, 26);
Group action invariants
Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_7^2:(C_3\times S_3)$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,21,2,18,4,19)(3,15,6,20,5,16)(7,17)(8,12,14)(10,13,11), (1,11,15)(2,10,16)(3,9,17)(4,8,18)(5,14,19)(6,13,20)(7,12,21) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 7: None
Low degree siblings
14T26, 21T25, 42T143, 42T144, 42T152, 42T153, 42T154, 42T155Siblings 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 $ | $1$ | $1$ | $()$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)$ |
$ 7, 7, 7 $ | $6$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,13,11, 9,14,12,10)(15,17,19,21,16,18,20)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$ |
$ 7, 7, 7 $ | $6$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,12, 9,13,10,14,11)(15,21,20,19,18,17,16)$ |
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $49$ | $3$ | $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$ |
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $49$ | $3$ | $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $98$ | $3$ | $( 1,21, 9)( 2,15, 8)( 3,16,14)( 4,17,13)( 5,18,12)( 6,19,11)( 7,20,10)$ |
$ 21 $ | $42$ | $21$ | $( 1,20,12, 2,15, 8, 3,17,11, 4,19,14, 5,21,10, 6,16,13, 7,18, 9)$ |
$ 21 $ | $42$ | $21$ | $( 1,20,10, 4,19,12, 7,18,14, 3,17, 9, 6,16,11, 2,15,13, 5,21, 8)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $14$ | $3$ | $( 1,20,13)( 2,15, 9)( 3,17,12)( 4,19, 8)( 5,21,11)( 6,16,14)( 7,18,10)$ |
$ 21 $ | $42$ | $21$ | $( 1,18,10, 4,16,11, 7,21,12, 3,19,13, 6,17,14, 2,15, 8, 5,20, 9)$ |
$ 21 $ | $42$ | $21$ | $( 1,18, 8, 3,19,11, 5,20,14, 7,21,10, 2,15,13, 4,16, 9, 6,17,12)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $14$ | $3$ | $( 1,18,11)( 2,15, 9)( 3,19,14)( 4,16,12)( 5,20,10)( 6,17, 8)( 7,21,13)$ |
$ 6, 6, 3, 3, 2, 1 $ | $147$ | $6$ | $( 2, 5, 3)( 4, 6, 7)( 8,18,13,19, 9,21)(10,17)(11,20,12,16,14,15)$ |
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $21$ | $2$ | $( 8,21)( 9,20)(10,19)(11,18)(12,17)(13,16)(14,15)$ |
$ 14, 7 $ | $63$ | $14$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,21,12,17, 9,20,13,16,10,19,14,15,11,18)$ |
$ 14, 7 $ | $63$ | $14$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,21,13,16,11,18, 9,20,14,15,12,17,10,19)$ |
$ 6, 6, 3, 3, 2, 1 $ | $147$ | $6$ | $( 2, 3, 5)( 4, 7, 6)( 8,20,10,16,11,21)( 9,18,14,15,13,17)(12,19)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $882=2 \cdot 3^{2} \cdot 7^{2}$ | 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: | 882.34 | magma: IdentifyGroup(G);
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Character table: |
2 1 . . 1 1 . 1 1 . . . . . . . 1 1 1 1 1 3 2 . 1 . . 1 2 2 2 1 1 2 1 1 2 1 1 . . 1 7 2 2 2 2 2 2 . . . 1 1 1 1 1 1 . 1 1 1 . 1a 7a 7b 7c 7d 7e 3a 3b 3c 21a 21b 3d 21c 21d 3e 6a 2a 14a 14b 6b 2P 1a 7a 7b 7c 7d 7e 3b 3a 3c 21d 21c 3e 21b 21a 3d 3a 1a 7d 7c 3b 3P 1a 7a 7e 7d 7c 7b 1a 1a 1a 7b 7e 1a 7e 7b 1a 2a 2a 14b 14a 2a 5P 1a 7a 7e 7d 7c 7b 3b 3a 3c 21c 21d 3e 21a 21b 3d 6b 2a 14b 14a 6a 7P 1a 1a 1a 1a 1a 1a 3a 3b 3c 3d 3d 3d 3e 3e 3e 6a 2a 2a 2a 6b 11P 1a 7a 7b 7c 7d 7e 3b 3a 3c 21d 21c 3e 21b 21a 3d 6b 2a 14a 14b 6a 13P 1a 7a 7e 7d 7c 7b 3a 3b 3c 21b 21a 3d 21d 21c 3e 6a 2a 14b 14a 6b 17P 1a 7a 7e 7d 7c 7b 3b 3a 3c 21c 21d 3e 21a 21b 3d 6b 2a 14b 14a 6a 19P 1a 7a 7e 7d 7c 7b 3a 3b 3c 21b 21a 3d 21d 21c 3e 6a 2a 14b 14a 6b 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 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 X.3 1 1 1 1 1 1 E /E 1 E E E /E /E /E -/E -1 -1 -1 -E X.4 1 1 1 1 1 1 /E E 1 /E /E /E E E E -E -1 -1 -1 -/E X.5 1 1 1 1 1 1 E /E 1 E E E /E /E /E /E 1 1 1 E X.6 1 1 1 1 1 1 /E E 1 /E /E /E E E E E 1 1 1 /E X.7 2 2 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 . . . . . X.8 2 2 2 2 2 2 F /F -1 -E -E -E -/E -/E -/E . . . . . X.9 2 2 2 2 2 2 /F F -1 -/E -/E -/E -E -E -E . . . . . X.10 6 -1 A C /C /A . . . G /G 3 /G G 3 . . . . . X.11 6 -1 /A /C C A . . . /G G 3 G /G 3 . . . . . X.12 6 -1 A C /C /A . . . H /I J /H I /J . . . . . X.13 6 -1 A C /C /A . . . I /H /J /I H J . . . . . X.14 6 -1 /A /C C A . . . /I H J I /H /J . . . . . X.15 6 -1 /A /C C A . . . /H I /J H /I J . . . . . X.16 9 2 B D /D /B . . . . . . . . . . -3 -G -/G . X.17 9 2 /B /D D B . . . . . . . . . . -3 -/G -G . X.18 9 2 B D /D /B . . . . . . . . . . 3 G /G . X.19 9 2 /B /D D B . . . . . . . . . . 3 /G G . X.20 18 -3 -3 4 4 -3 . . . . . . . . . . . . . . A = -2*E(7)-2*E(7)^2-3*E(7)^3-2*E(7)^4-3*E(7)^5-3*E(7)^6 = (5+Sqrt(-7))/2 = 3+b7 B = 3*E(7)^3+3*E(7)^5+3*E(7)^6 = (-3-3*Sqrt(-7))/2 = -3-3b7 C = 2*E(7)^3+2*E(7)^5+2*E(7)^6 = -1-Sqrt(-7) = -1-i7 D = E(7)+E(7)^2+2*E(7)^3+E(7)^4+2*E(7)^5+2*E(7)^6 = (-3-Sqrt(-7))/2 = -2-b7 E = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 F = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3 G = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7 H = E(21)^2+E(21)^8+E(21)^11 I = E(21)+E(21)^4+E(21)^16 J = 3*E(3)^2 = (-3-3*Sqrt(-3))/2 = -3-3b3 |
magma: CharacterTable(G);