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Magma
magma: G := TransitiveGroup(20, 44);
Group action invariants
Degree $n$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $44$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2\wr C_5$ | ||
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: | (1,5,9,14,18)(2,6,10,13,17)(3,7,12,16,20)(4,8,11,15,19), (1,2)(3,4)(5,6)(7,8)(9,19)(10,20)(11,12)(13,14)(15,16)(17,18) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $5$: $C_5$ $10$: $C_{10}$ $80$: $C_2^4 : C_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: None
Degree 5: $C_5$
Degree 10: $C_{10}$, $C_2^4 : C_5$, $C_2 \times (C_2^4 : C_5)$
Low degree siblings
10T14 x 3, 20T40 x 12, 20T41 x 6, 20T44 x 2, 20T46 x 3, 32T2133, 40T121 x 6, 40T122 x 6, 40T123 x 12, 40T141, 40T142 x 3Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 7,18)( 8,17)( 9,20)(10,19)$ | |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 5,16)( 6,15)( 9,20)(10,19)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $5$ | $2$ | $( 3,14)( 4,13)( 5,16)( 6,15)( 7,18)( 8,17)( 9,20)(10,19)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,19)(10,20)(11,12)(13,14)(15,16)(17,18)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 2)( 3, 4)( 5,15)( 6,16)( 7,17)( 8,18)( 9,19)(10,20)(11,12)(13,14)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 2)( 3,13)( 4,14)( 5, 6)( 7,17)( 8,18)( 9,19)(10,20)(11,12)(15,16)$ | |
$ 5, 5, 5, 5 $ | $16$ | $5$ | $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)(11,13,15,17,19)(12,14,16,18,20)$ | |
$ 10, 10 $ | $16$ | $10$ | $( 1, 4, 5, 8, 9,11,14,15,18,19)( 2, 3, 6, 7,10,12,13,16,17,20)$ | |
$ 5, 5, 5, 5 $ | $16$ | $5$ | $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)(11,15,19,13,17)(12,16,20,14,18)$ | |
$ 10, 10 $ | $16$ | $10$ | $( 1, 6, 9,13,18,11,16,19, 3, 8)( 2, 5,10,14,17,12,15,20, 4, 7)$ | |
$ 5, 5, 5, 5 $ | $16$ | $5$ | $( 1, 7, 3, 9, 5)( 2, 8, 4,10, 6)(11,17,13,19,15)(12,18,14,20,16)$ | |
$ 10, 10 $ | $16$ | $10$ | $( 1, 8, 3,10,16,11,18,13,20, 6)( 2, 7, 4, 9,15,12,17,14,19, 5)$ | |
$ 5, 5, 5, 5 $ | $16$ | $5$ | $( 1, 9, 7, 5, 3)( 2,10, 8, 6, 4)(11,19,17,15,13)(12,20,18,16,14)$ | |
$ 10, 10 $ | $16$ | $10$ | $( 1,10,18,15,14,11,20, 8, 5, 4)( 2, 9,17,16,13,12,19, 7, 6, 3)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,11)( 2,12)( 3,13)( 4,14)( 5,15)( 6,16)( 7,17)( 8,18)( 9,19)(10,20)$ |
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: | 160.235 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 5A1 | 5A-1 | 5A2 | 5A-2 | 10A1 | 10A-1 | 10A3 | 10A-3 | ||
Size | 1 | 1 | 5 | 5 | 5 | 5 | 5 | 5 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | |
2 P | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 5A2 | 5A-2 | 5A-1 | 5A1 | 5A-1 | 5A-2 | 5A1 | 5A2 | |
5 P | 1A | 2A | 2F | 2B | 2D | 2C | 2G | 2E | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | |
Type | |||||||||||||||||
160.235.1a | R | ||||||||||||||||
160.235.1b | R | ||||||||||||||||
160.235.1c1 | C | ||||||||||||||||
160.235.1c2 | C | ||||||||||||||||
160.235.1c3 | C | ||||||||||||||||
160.235.1c4 | C | ||||||||||||||||
160.235.1d1 | C | ||||||||||||||||
160.235.1d2 | C | ||||||||||||||||
160.235.1d3 | C | ||||||||||||||||
160.235.1d4 | C | ||||||||||||||||
160.235.5a | R | ||||||||||||||||
160.235.5b | R | ||||||||||||||||
160.235.5c | R | ||||||||||||||||
160.235.5d | R | ||||||||||||||||
160.235.5e | R | ||||||||||||||||
160.235.5f | R |
magma: CharacterTable(G);