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Magma
magma: G := TransitiveGroup(18, 32);
Group action invariants
Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $S_3\times A_4$ | ||
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: | (1,12,14)(2,11,13)(3,7,16)(4,8,15)(5,9,17)(6,10,18), (1,5)(2,6)(7,10)(8,9)(11,12)(13,15)(14,16) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $12$: $A_4$ $18$: $S_3\times C_3$ $24$: $A_4\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 6: $A_4\times C_2$
Degree 9: $S_3\times C_3$
Low degree siblings
12T43, 18T31, 24T78, 24T83, 36T21, 36T50, 36T51Siblings 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 $ | $3$ | $2$ | $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $9$ | $2$ | $( 3, 5)( 4, 6)( 7,11)( 8,12)( 9,10)(13,18)(14,17)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 6)( 4, 5)( 7,11)( 8,12)( 9,10)(13,17)(14,18)(15,16)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7, 9,12)( 8,10,11)(13,15,18)(14,16,17)$ | |
$ 6, 6, 3, 3 $ | $6$ | $6$ | $( 1, 3, 5)( 2, 4, 6)( 7,10,12, 8, 9,11)(13,16,18,14,15,17)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 7,17)( 2, 8,18)( 3, 9,14)( 4,10,13)( 5,12,16)( 6,11,15)$ | |
$ 6, 6, 6 $ | $12$ | $6$ | $( 1, 7,15, 6,10,14)( 2, 8,16, 5, 9,13)( 3,12,18, 4,11,17)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 9,15)( 2,10,16)( 3,12,18)( 4,11,17)( 5, 7,13)( 6, 8,14)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1,13,12)( 2,14,11)( 3,15, 7)( 4,16, 8)( 5,18, 9)( 6,17,10)$ | |
$ 6, 6, 6 $ | $12$ | $6$ | $( 1,13,10, 6,16, 7)( 2,14, 9, 5,15, 8)( 3,18,11, 4,17,12)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,15, 9)( 2,16,10)( 3,18,12)( 4,17,11)( 5,13, 7)( 6,14, 8)$ |
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: | 72.44 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 3A | 3B1 | 3B-1 | 3C1 | 3C-1 | 6A | 6B1 | 6B-1 | ||
Size | 1 | 3 | 3 | 9 | 2 | 4 | 4 | 8 | 8 | 6 | 12 | 12 | |
2 P | 1A | 1A | 1A | 1A | 3A | 3B-1 | 3B1 | 3C-1 | 3C1 | 3A | 3B1 | 3B-1 | |
3 P | 1A | 2A | 2B | 2C | 1A | 1A | 1A | 1A | 1A | 2A | 2B | 2B | |
Type | |||||||||||||
72.44.1a | R | ||||||||||||
72.44.1b | R | ||||||||||||
72.44.1c1 | C | ||||||||||||
72.44.1c2 | C | ||||||||||||
72.44.1d1 | C | ||||||||||||
72.44.1d2 | C | ||||||||||||
72.44.2a | R | ||||||||||||
72.44.2b1 | C | ||||||||||||
72.44.2b2 | C | ||||||||||||
72.44.3a | R | ||||||||||||
72.44.3b | R | ||||||||||||
72.44.6a | R |
magma: CharacterTable(G);