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Magma
magma: G := TransitiveGroup(18, 34);
Group action invariants
Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $34$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $\SOPlus(4,2)$ | ||
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: | (1,12,10)(2,11,9)(3,16,6,17,14,7)(4,15,5,18,13,8), (1,15,3,8)(2,16,4,7)(5,10,11,14)(6,9,12,13)(17,18) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $D_{4}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 6: $C_3^2:D_4$
Degree 9: $S_3^2:C_2$
Low degree siblings
6T13 x 2, 9T16, 12T34 x 2, 12T35 x 2, 12T36 x 2, 18T34, 18T36, 24T72 x 2, 36T53, 36T54 x 2Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 5,13)( 6,14)( 7,16)( 8,15)( 9,11)(10,12)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $9$ | $2$ | $( 3,17)( 4,18)( 5,15)( 6,16)( 7,14)( 8,13)( 9,11)(10,12)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 2)( 3, 9)( 4,10)( 5, 6)( 7,13)( 8,14)(11,17)(12,18)(15,16)$ |
$ 4, 4, 4, 4, 2 $ | $18$ | $4$ | $( 1, 2)( 3, 9,17,11)( 4,10,18,12)( 5, 7,15,14)( 6, 8,16,13)$ |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 3,17)( 2, 4,18)( 5, 8, 9)( 6, 7,10)(11,13,15)(12,14,16)$ |
$ 6, 6, 3, 3 $ | $12$ | $6$ | $( 1, 3,17)( 2, 4,18)( 5,15, 9,13, 8,11)( 6,16,10,14, 7,12)$ |
$ 6, 6, 6 $ | $12$ | $6$ | $( 1, 4, 6, 8,16,11)( 2, 3, 5, 7,15,12)( 9,17,13,10,18,14)$ |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 6,16)( 2, 5,15)( 3, 7,12)( 4, 8,11)( 9,13,18)(10,14,17)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $72=2^{3} \cdot 3^{2}$ | 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: | 72.40 | magma: IdentifyGroup(G);
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Character table: |
2 3 2 3 2 2 1 1 1 1 3 2 1 . 1 . 2 1 1 2 1a 2a 2b 2c 4a 3a 6a 6b 3b 2P 1a 1a 1a 1a 2b 3a 3a 3b 3b 3P 1a 2a 2b 2c 4a 1a 2a 2c 1a 5P 1a 2a 2b 2c 4a 3a 6a 6b 3b X.1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 1 -1 -1 1 X.3 1 -1 1 1 -1 1 -1 1 1 X.4 1 1 1 -1 -1 1 1 -1 1 X.5 2 . -2 . . 2 . . 2 X.6 4 -2 . . . 1 1 . -2 X.7 4 . . -2 . -2 . 1 1 X.8 4 . . 2 . -2 . -1 1 X.9 4 2 . . . 1 -1 . -2 |
magma: CharacterTable(G);