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Magma
magma: G := TransitiveGroup(20, 39);
Group action invariants
Degree $n$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2^4:D_5$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,9,12,20)(2,10,11,19)(3,18,13,8)(4,17,14,7)(5,6)(15,16), (1,3,11,13)(2,4,12,14)(5,20,6,19)(7,18)(8,17)(9,15,10,16) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $10$: $D_{5}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: None
Degree 5: $D_{5}$
Degree 10: $(C_2^4 : C_5) : C_2$ x 3
Low degree siblings
10T15 x 3, 10T16 x 3, 16T415, 20T38 x 6, 20T43 x 3, 20T45 x 3, 32T2132, 40T143 x 3, 40T144 x 3, 40T145 x 6, 40T146Siblings 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 $ | $1$ | $1$ | $()$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $5$ | $2$ | $( 3, 4)( 5,15)( 6,16)( 7,17)( 8,18)( 9,10)(13,14)(19,20)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $20$ | $2$ | $( 3, 9)( 4,10)( 5, 7)( 6, 8)(13,19)(14,20)(15,17)(16,18)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $5$ | $2$ | $( 3,13)( 4,14)( 5,16)( 6,15)( 7,18)( 8,17)( 9,19)(10,20)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $5$ | $2$ | $( 3,14)( 4,13)( 5, 6)( 7, 8)( 9,20)(10,19)(15,16)(17,18)$ | |
$ 4, 4, 4, 4, 2, 2 $ | $20$ | $4$ | $( 1, 2)( 3, 9,13,19)( 4,10,14,20)( 5, 8,16,17)( 6, 7,15,18)(11,12)$ | |
$ 4, 4, 4, 4, 2, 2 $ | $20$ | $4$ | $( 1, 3, 2, 4)( 5,19,16,10)( 6,20,15, 9)( 7,17)( 8,18)(11,13,12,14)$ | |
$ 5, 5, 5, 5 $ | $32$ | $5$ | $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)(11,13,15,17,19)(12,14,16,18,20)$ | |
$ 4, 4, 4, 4, 2, 2 $ | $20$ | $4$ | $( 1, 3,11,13)( 2, 4,12,14)( 5,20, 6,19)( 7,18)( 8,17)( 9,15,10,16)$ | |
$ 5, 5, 5, 5 $ | $32$ | $5$ | $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)(11,15,19,13,17)(12,16,20,14,18)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $160=2^{5} \cdot 5$ | 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: | 160.234 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 5A1 | 5A2 | ||
Size | 1 | 5 | 5 | 5 | 20 | 20 | 20 | 20 | 32 | 32 | |
2 P | 1A | 1A | 1A | 1A | 1A | 2A | 2B | 2C | 5A2 | 5A1 | |
5 P | 1A | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 1A | 1A | |
Type | |||||||||||
160.234.1a | R | ||||||||||
160.234.1b | R | ||||||||||
160.234.2a1 | R | ||||||||||
160.234.2a2 | R | ||||||||||
160.234.5a | R | ||||||||||
160.234.5b | R | ||||||||||
160.234.5c | R | ||||||||||
160.234.5d | R | ||||||||||
160.234.5e | R | ||||||||||
160.234.5f | R |
magma: CharacterTable(G);