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Magma
magma: G := TransitiveGroup(18, 26);
Group action invariants
Degree $n$: | $18$ | 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: | $C_2^2:C_{18}$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,7,16,6,11,13,3,10,17)(2,8,15,5,12,14,4,9,18), (1,8,15,5,11,13,3,9,18,2,7,16,6,12,14,4,10,17) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $9$: $C_9$ $12$: $A_4$ $18$: $C_{18}$ $24$: $A_4\times C_2$ $36$: $C_2^2 : C_9$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 6: $A_4\times C_2$
Degree 9: $C_9$
Low degree siblings
36T16, 36T30Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $(13,14)(15,16)(17,18)$ | |
$ 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, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ | |
$ 6, 6, 3, 3 $ | $3$ | $6$ | $( 1, 3, 6)( 2, 4, 5)( 7, 9,11, 8,10,12)(13,15,17,14,16,18)$ | |
$ 6, 3, 3, 3, 3 $ | $3$ | $6$ | $( 1, 3, 6)( 2, 4, 5)( 7, 9,11, 8,10,12)(13,16,17)(14,15,18)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 6)( 2, 4, 5)( 7,10,11)( 8, 9,12)(13,16,17)(14,15,18)$ | |
$ 6, 6, 6 $ | $1$ | $6$ | $( 1, 4, 6, 2, 3, 5)( 7, 9,11, 8,10,12)(13,15,17,14,16,18)$ | |
$ 6, 3, 3, 3, 3 $ | $3$ | $6$ | $( 1, 5, 3, 2, 6, 4)( 7,11,10)( 8,12, 9)(13,17,16)(14,18,15)$ | |
$ 6, 6, 3, 3 $ | $3$ | $6$ | $( 1, 5, 3, 2, 6, 4)( 7,11,10)( 8,12, 9)(13,18,16,14,17,15)$ | |
$ 6, 6, 6 $ | $1$ | $6$ | $( 1, 5, 3, 2, 6, 4)( 7,12,10, 8,11, 9)(13,18,16,14,17,15)$ | |
$ 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 6, 3)( 2, 5, 4)( 7,11,10)( 8,12, 9)(13,17,16)(14,18,15)$ | |
$ 18 $ | $4$ | $18$ | $( 1, 7,15, 5,12,13, 3,10,18, 2, 8,16, 6,11,14, 4, 9,17)$ | |
$ 9, 9 $ | $4$ | $9$ | $( 1, 7,15, 6,11,14, 3,10,18)( 2, 8,16, 5,12,13, 4, 9,17)$ | |
$ 9, 9 $ | $4$ | $9$ | $( 1, 9,14, 6, 8,18, 3,12,15)( 2,10,13, 5, 7,17, 4,11,16)$ | |
$ 18 $ | $4$ | $18$ | $( 1, 9,14, 5, 7,17, 3,12,15, 2,10,13, 6, 8,18, 4,11,16)$ | |
$ 9, 9 $ | $4$ | $9$ | $( 1,11,17, 6,10,16, 3, 7,13)( 2,12,18, 5, 9,15, 4, 8,14)$ | |
$ 18 $ | $4$ | $18$ | $( 1,11,17, 5, 9,15, 3, 7,13, 2,12,18, 6,10,16, 4, 8,14)$ | |
$ 9, 9 $ | $4$ | $9$ | $( 1,13, 7, 3,16,10, 6,17,11)( 2,14, 8, 4,15, 9, 5,18,12)$ | |
$ 18 $ | $4$ | $18$ | $( 1,13, 8, 4,15,10, 6,17,12, 2,14, 7, 3,16, 9, 5,18,11)$ | |
$ 18 $ | $4$ | $18$ | $( 1,15,11, 4,17, 8, 6,14,10, 2,16,12, 3,18, 7, 5,13, 9)$ | |
$ 9, 9 $ | $4$ | $9$ | $( 1,15,12, 3,18, 8, 6,14, 9)( 2,16,11, 4,17, 7, 5,13,10)$ | |
$ 9, 9 $ | $4$ | $9$ | $( 1,17,10, 3,13,11, 6,16, 7)( 2,18, 9, 4,14,12, 5,15, 8)$ | |
$ 18 $ | $4$ | $18$ | $( 1,17, 9, 4,14,11, 6,16, 8, 2,18,10, 3,13,12, 5,15, 7)$ |
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: | 72.16 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 3A1 | 3A-1 | 6A1 | 6A-1 | 6B1 | 6B-1 | 6C1 | 6C-1 | 9A1 | 9A-1 | 9A2 | 9A-2 | 9A4 | 9A-4 | 18A1 | 18A-1 | 18A5 | 18A-5 | 18A7 | 18A-7 | ||
Size | 1 | 1 | 3 | 3 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
2 P | 1A | 1A | 1A | 1A | 3A-1 | 3A1 | 3A1 | 3A-1 | 3A-1 | 3A1 | 3A-1 | 3A1 | 9A1 | 9A-2 | 9A-4 | 9A2 | 9A4 | 9A-1 | 9A4 | 9A-1 | 9A-4 | 9A-2 | 9A1 | 9A2 | |
3 P | 1A | 2A | 2B | 2C | 1A | 1A | 2A | 2A | 2B | 2C | 2C | 2B | 3A-1 | 3A-1 | 3A1 | 3A1 | 3A-1 | 3A1 | 6A1 | 6A-1 | 6A-1 | 6A1 | 6A1 | 6A-1 | |
Type | |||||||||||||||||||||||||
72.16.1a | R | ||||||||||||||||||||||||
72.16.1b | R | ||||||||||||||||||||||||
72.16.1c1 | C | ||||||||||||||||||||||||
72.16.1c2 | C | ||||||||||||||||||||||||
72.16.1d1 | C | ||||||||||||||||||||||||
72.16.1d2 | C | ||||||||||||||||||||||||
72.16.1e1 | C | ||||||||||||||||||||||||
72.16.1e2 | C | ||||||||||||||||||||||||
72.16.1e3 | C | ||||||||||||||||||||||||
72.16.1e4 | C | ||||||||||||||||||||||||
72.16.1e5 | C | ||||||||||||||||||||||||
72.16.1e6 | C | ||||||||||||||||||||||||
72.16.1f1 | C | ||||||||||||||||||||||||
72.16.1f2 | C | ||||||||||||||||||||||||
72.16.1f3 | C | ||||||||||||||||||||||||
72.16.1f4 | C | ||||||||||||||||||||||||
72.16.1f5 | C | ||||||||||||||||||||||||
72.16.1f6 | C | ||||||||||||||||||||||||
72.16.3a | R | ||||||||||||||||||||||||
72.16.3b | R | ||||||||||||||||||||||||
72.16.3c1 | C | ||||||||||||||||||||||||
72.16.3c2 | C | ||||||||||||||||||||||||
72.16.3d1 | C | ||||||||||||||||||||||||
72.16.3d2 | C |
magma: CharacterTable(G);