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Magma
magma: G := TransitiveGroup(18, 43);
Group action invariants
Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $43$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3\times S_3^2$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,12,13,4,8,18)(2,10,14,5,9,16)(3,11,15,6,7,17), (1,8,2,9,3,7)(4,16,5,17,6,18)(10,12,11)(13,15,14) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $S_3$ x 2, $C_6$ x 3 $12$: $D_{6}$ x 2, $C_6\times C_2$ $18$: $S_3\times C_3$ x 2 $36$: $S_3^2$, $C_6\times S_3$ x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 9: None
Low degree siblings
12T70, 18T46 x 2, 27T36, 36T80, 36T82 x 2, 36T92Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | 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 $ | $9$ | $2$ | $( 7,13)( 8,14)( 9,15)(10,17)(11,18)(12,16)$ | |
$ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $3$ | $( 4,10,17)( 5,11,18)( 6,12,16)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$ | |
$ 6, 6, 3, 3 $ | $9$ | $6$ | $( 1, 2, 3)( 4, 5, 6)( 7,14, 9,13, 8,15)(10,18,12,17,11,16)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 2, 3)( 4,11,16)( 5,12,17)( 6,10,18)( 7, 8, 9)(13,14,15)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)$ | |
$ 6, 6, 3, 3 $ | $9$ | $6$ | $( 1, 3, 2)( 4, 6, 5)( 7,15, 8,13, 9,14)(10,16,11,17,12,18)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 3, 2)( 4,12,18)( 5,10,16)( 6,11,17)( 7, 9, 8)(13,15,14)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 4)( 2, 5)( 3, 6)( 7,11)( 8,12)( 9,10)(13,18)(14,16)(15,17)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 4)( 2, 5)( 3, 6)( 7,18)( 8,16)( 9,17)(10,15)(11,13)(12,14)$ | |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1, 4, 9,10,15,17)( 2, 5, 7,11,13,18)( 3, 6, 8,12,14,16)$ | |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1, 4, 9,17,15,10)( 2, 5, 7,18,13,11)( 3, 6, 8,16,14,12)$ | |
$ 6, 6, 6 $ | $3$ | $6$ | $( 1, 5, 3, 4, 2, 6)( 7,12, 9,11, 8,10)(13,16,15,18,14,17)$ | |
$ 6, 6, 6 $ | $3$ | $6$ | $( 1, 5, 3, 4, 2, 6)( 7,16, 9,18, 8,17)(10,13,12,15,11,14)$ | |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1, 5, 8,10,13,16)( 2, 6, 9,11,14,17)( 3, 4, 7,12,15,18)$ | |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1, 5, 8,17,13,12)( 2, 6, 9,18,14,10)( 3, 4, 7,16,15,11)$ | |
$ 6, 6, 6 $ | $3$ | $6$ | $( 1, 6, 2, 4, 3, 5)( 7,10, 8,11, 9,12)(13,17,14,18,15,16)$ | |
$ 6, 6, 6 $ | $3$ | $6$ | $( 1, 6, 2, 4, 3, 5)( 7,17, 8,18, 9,16)(10,14,11,15,12,13)$ | |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1, 6, 7,10,14,18)( 2, 4, 8,11,15,16)( 3, 5, 9,12,13,17)$ | |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1, 6, 7,17,14,11)( 2, 4, 8,18,15,12)( 3, 5, 9,16,13,10)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 7,14)( 2, 8,15)( 3, 9,13)( 4,11,16)( 5,12,17)( 6,10,18)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 7,14)( 2, 8,15)( 3, 9,13)( 4,18,12)( 5,16,10)( 6,17,11)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 8,13)( 2, 9,14)( 3, 7,15)( 4,12,18)( 5,10,16)( 6,11,17)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 8,13)( 2, 9,14)( 3, 7,15)( 4,16,11)( 5,17,12)( 6,18,10)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 9,15)( 2, 7,13)( 3, 8,14)( 4,10,17)( 5,11,18)( 6,12,16)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 9,15)( 2, 7,13)( 3, 8,14)( 4,17,10)( 5,18,11)( 6,16,12)$ |
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: | 108.38 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 3A1 | 3A-1 | 3B | 3C | 3D1 | 3D-1 | 3E1 | 3E-1 | 3F | 3G1 | 3G-1 | 6A1 | 6A-1 | 6B1 | 6B-1 | 6C | 6D | 6E1 | 6E-1 | 6F1 | 6F-1 | 6G1 | 6G-1 | ||
Size | 1 | 3 | 3 | 9 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | |
2 P | 1A | 1A | 1A | 1A | 3A-1 | 3A1 | 3D-1 | 3C | 3E1 | 3D1 | 3E-1 | 3B | 3F | 3G-1 | 3G1 | 3A1 | 3A-1 | 3A1 | 3A-1 | 3B | 3D1 | 3D-1 | 3C | 3E1 | 3E-1 | 3A1 | 3A-1 | |
3 P | 1A | 2A | 2B | 2C | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2B | 2B | 2A | 2A | 2A | 2A | 2A | 2B | 2B | 2B | 2C | 2C | |
Type | ||||||||||||||||||||||||||||
108.38.1a | R | |||||||||||||||||||||||||||
108.38.1b | R | |||||||||||||||||||||||||||
108.38.1c | R | |||||||||||||||||||||||||||
108.38.1d | R | |||||||||||||||||||||||||||
108.38.1e1 | C | |||||||||||||||||||||||||||
108.38.1e2 | C | |||||||||||||||||||||||||||
108.38.1f1 | C | |||||||||||||||||||||||||||
108.38.1f2 | C | |||||||||||||||||||||||||||
108.38.1g1 | C | |||||||||||||||||||||||||||
108.38.1g2 | C | |||||||||||||||||||||||||||
108.38.1h1 | C | |||||||||||||||||||||||||||
108.38.1h2 | C | |||||||||||||||||||||||||||
108.38.2a | R | |||||||||||||||||||||||||||
108.38.2b | R | |||||||||||||||||||||||||||
108.38.2c | R | |||||||||||||||||||||||||||
108.38.2d | R | |||||||||||||||||||||||||||
108.38.2e1 | C | |||||||||||||||||||||||||||
108.38.2e2 | C | |||||||||||||||||||||||||||
108.38.2f1 | C | |||||||||||||||||||||||||||
108.38.2f2 | C | |||||||||||||||||||||||||||
108.38.2g1 | C | |||||||||||||||||||||||||||
108.38.2g2 | C | |||||||||||||||||||||||||||
108.38.2h1 | C | |||||||||||||||||||||||||||
108.38.2h2 | C | |||||||||||||||||||||||||||
108.38.4a | R | |||||||||||||||||||||||||||
108.38.4b1 | C | |||||||||||||||||||||||||||
108.38.4b2 | C |
magma: CharacterTable(G);