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Magma
magma: G := TransitiveGroup(18, 42);
Group action invariants
Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3^2:D_6$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,2)(3,4)(5,7,10,6,8,9)(11,15,13,12,16,14)(17,18), (1,5,16,2,6,15)(3,9,12,17,8,13)(4,10,11,18,7,14) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $S_3$, $C_6$ x 3 $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$ $36$: $C_6\times S_3$ $54$: $C_3^2 : C_6$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 6: $C_6$
Degree 9: $C_3^2 : S_3 $
Low degree siblings
18T41 x 2, 18T42, 36T71, 36T73, 36T75Siblings 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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $6$ | $3$ | $( 5, 8,10)( 6, 7, 9)(11,16,13)(12,15,14)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 3,18)( 4,17)( 7, 9)( 8,10)(11,13)(12,14)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ |
$ 6, 6, 2, 2, 2 $ | $6$ | $6$ | $( 1, 2)( 3, 4)( 5, 7,10, 6, 8, 9)(11,15,13,12,16,14)(17,18)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $9$ | $2$ | $( 1, 2)( 3,17)( 4,18)( 5, 6)( 7,10)( 8, 9)(11,14)(12,13)(15,16)$ |
$ 6, 6, 6 $ | $2$ | $6$ | $( 1, 3,17, 2, 4,18)( 5, 7,10, 6, 8, 9)(11,14,16,12,13,15)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 4,17)( 2, 3,18)( 5, 8,10)( 6, 7, 9)(11,13,16)(12,14,15)$ |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1, 5,11, 2, 6,12)( 3, 7,14, 4, 8,13)( 9,15,17,10,16,18)$ |
$ 6, 6, 6 $ | $3$ | $6$ | $( 1, 5,16, 2, 6,15)( 3, 7,12, 4, 8,11)( 9,14,17,10,13,18)$ |
$ 6, 6, 6 $ | $9$ | $6$ | $( 1, 5,11, 3, 9,14)( 2, 6,12, 4,10,13)( 7,15,17, 8,16,18)$ |
$ 3, 3, 3, 3, 3, 3 $ | $6$ | $3$ | $( 1, 6,11)( 2, 5,12)( 3, 8,14)( 4, 7,13)( 9,16,17)(10,15,18)$ |
$ 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1, 6,16)( 2, 5,15)( 3, 8,12)( 4, 7,11)( 9,13,17)(10,14,18)$ |
$ 6, 6, 3, 3 $ | $9$ | $6$ | $( 1, 6,11, 4, 9,13)( 2, 5,12, 3,10,14)( 7,16,17)( 8,15,18)$ |
$ 3, 3, 3, 3, 3, 3 $ | $6$ | $3$ | $( 1,11, 6)( 2,12, 5)( 3,14, 8)( 4,13, 7)( 9,17,16)(10,18,15)$ |
$ 3, 3, 3, 3, 3, 3 $ | $3$ | $3$ | $( 1,11, 9)( 2,12,10)( 3,14, 5)( 4,13, 6)( 7,17,16)( 8,18,15)$ |
$ 6, 6, 3, 3 $ | $9$ | $6$ | $( 1,11, 7,17,13, 6)( 2,12, 8,18,14, 5)( 3,15,10)( 4,16, 9)$ |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1,12, 6, 2,11, 5)( 3,13, 8, 4,14, 7)( 9,18,16,10,17,15)$ |
$ 6, 6, 6 $ | $3$ | $6$ | $( 1,12, 9, 2,11,10)( 3,13, 5, 4,14, 6)( 7,18,16, 8,17,15)$ |
$ 6, 6, 6 $ | $9$ | $6$ | $( 1,12, 7,18,13, 5)( 2,11, 8,17,14, 6)( 3,16,10, 4,15, 9)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $108=2^{2} \cdot 3^{3}$ | 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: | 108.25 | magma: IdentifyGroup(G);
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Character table: |
2 2 1 2 2 1 2 1 1 1 2 2 1 2 2 1 2 2 1 2 3 3 2 1 3 2 1 3 3 2 2 1 2 2 1 2 2 1 2 2 1a 3a 2a 2b 6a 2c 6b 3b 6c 6d 6e 3c 3d 6f 3e 3f 6g 6h 6i 2P 1a 3a 1a 1a 3a 1a 3b 3b 3e 3f 3f 3e 3f 3f 3c 3d 3d 3c 3d 3P 1a 1a 2a 2b 2b 2c 2b 1a 2b 2b 2c 1a 1a 2a 1a 1a 2a 2b 2b 5P 1a 3a 2a 2b 6a 2c 6b 3b 6h 6i 6j 3e 3f 6g 3c 3d 6f 6c 6d 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 1 1 -1 1 1 1 1 -1 1 1 -1 1 1 -1 1 1 X.4 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 1 1 1 1 1 1 -1 -1 X.5 1 1 -1 -1 -1 1 -1 1 A A -A -A -A A -/A -/A /A /A /A X.6 1 1 -1 -1 -1 1 -1 1 /A /A -/A -/A -/A /A -A -A A A A X.7 1 1 -1 1 1 -1 1 1 -/A -/A /A -/A -/A /A -A -A A -A -A X.8 1 1 -1 1 1 -1 1 1 -A -A A -A -A A -/A -/A /A -/A -/A X.9 1 1 1 -1 -1 -1 -1 1 A A A -A -A -A -/A -/A -/A /A /A X.10 1 1 1 -1 -1 -1 -1 1 /A /A /A -/A -/A -/A -A -A -A A A X.11 1 1 1 1 1 1 1 1 -/A -/A -/A -/A -/A -/A -A -A -A -A -A X.12 1 1 1 1 1 1 1 1 -A -A -A -A -A -A -/A -/A -/A -/A -/A X.13 2 -1 . 2 -1 . 2 2 -1 2 . -1 2 . -1 2 . -1 2 X.14 2 -1 . -2 1 . -2 2 1 -2 . -1 2 . -1 2 . 1 -2 X.15 2 -1 . 2 -1 . 2 2 A B . A B . /A /B . /A /B X.16 2 -1 . 2 -1 . 2 2 /A /B . /A /B . A B . A B X.17 2 -1 . -2 1 . -2 2 -A -B . A B . /A /B . -/A -/B X.18 2 -1 . -2 1 . -2 2 -/A -/B . /A /B . A B . -A -B X.19 6 . . -6 . . 3 -3 . . . . . . . . . . . X.20 6 . . 6 . . -3 -3 . . . . . . . . . . . 2 2 3 1 6j 2P 3d 3P 2c 5P 6e X.1 1 X.2 1 X.3 -1 X.4 -1 X.5 -/A X.6 -A X.7 A X.8 /A X.9 /A X.10 A X.11 -A X.12 -/A X.13 . X.14 . X.15 . X.16 . X.17 . X.18 . X.19 . X.20 . A = -E(3) = (1-Sqrt(-3))/2 = -b3 B = 2*E(3) = -1+Sqrt(-3) = 2b3 |
magma: CharacterTable(G);