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Magma
magma: G := TransitiveGroup(20, 81);
Group action invariants
Degree $n$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $81$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2\wr D_5$ | ||
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: | (1,6,19,14,8,11,16,9,4,18)(2,5,20,13,7,12,15,10,3,17), (1,3,11,13)(2,4,12,14)(5,9,15,19)(6,10,16,20)(7,18)(8,17) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $10$: $D_{5}$ $20$: $D_{10}$ $160$: $(C_2^4 : C_5) : C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: None
Degree 5: $D_{5}$
Degree 10: $D_5$, $C_2\times (C_2^4 : D_5)$ x 2
Low degree siblings
10T23 x 6, 20T71 x 6, 20T73 x 6, 20T76 x 6, 20T81 x 2, 20T85 x 6, 20T87 x 6, 32T9313 x 2, 40T204 x 3, 40T270 x 12, 40T271 x 12, 40T272 x 3, 40T273 x 2, 40T284 x 6, 40T286 x 6, 40T288 x 3, 40T293 x 3, 40T295 x 6Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 9,19)(10,20)$ | |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 7,17)( 8,18)( 9,19)(10,20)$ | |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 5,15)( 6,16)( 9,19)(10,20)$ | |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 5,15)( 6,16)( 7,17)( 8,18)( 9,19)(10,20)$ | |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 3,13)( 4,14)( 7,17)( 8,18)( 9,19)(10,20)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $5$ | $2$ | $( 3,13)( 4,14)( 5,15)( 6,16)( 7,17)( 8,18)( 9,19)(10,20)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $20$ | $2$ | $( 1, 2)( 3, 9)( 4,10)( 5, 8)( 6, 7)(11,12)(13,19)(14,20)(15,18)(16,17)$ | |
$ 4, 4, 2, 2, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1, 2)( 3, 9,13,19)( 4,10,14,20)( 5, 8)( 6, 7)(11,12)(15,18)(16,17)$ | |
$ 4, 4, 2, 2, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1, 2)( 3, 9)( 4,10)( 5, 8,15,18)( 6, 7,16,17)(11,12)(13,19)(14,20)$ | |
$ 4, 4, 4, 4, 2, 2 $ | $20$ | $4$ | $( 1, 2)( 3, 9,13,19)( 4,10,14,20)( 5, 8,15,18)( 6, 7,16,17)(11,12)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $20$ | $2$ | $( 1, 3)( 2, 4)( 5, 9)( 6,10)( 7,18)( 8,17)(11,13)(12,14)(15,19)(16,20)$ | |
$ 4, 4, 2, 2, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1, 3)( 2, 4)( 5, 9,15,19)( 6,10,16,20)( 7,18)( 8,17)(11,13)(12,14)$ | |
$ 4, 4, 2, 2, 2, 2, 2, 2 $ | $20$ | $4$ | $( 1, 3,11,13)( 2, 4,12,14)( 5, 9)( 6,10)( 7,18)( 8,17)(15,19)(16,20)$ | |
$ 4, 4, 4, 4, 2, 2 $ | $20$ | $4$ | $( 1, 3,11,13)( 2, 4,12,14)( 5, 9,15,19)( 6,10,16,20)( 7,18)( 8,17)$ | |
$ 5, 5, 5, 5 $ | $32$ | $5$ | $( 1, 4, 6, 8, 9)( 2, 3, 5, 7,10)(11,14,16,18,19)(12,13,15,17,20)$ | |
$ 10, 10 $ | $32$ | $10$ | $( 1, 4, 6, 8, 9,11,14,16,18,19)( 2, 3, 5, 7,10,12,13,15,17,20)$ | |
$ 5, 5, 5, 5 $ | $32$ | $5$ | $( 1, 6, 9, 4, 8)( 2, 5,10, 3, 7)(11,16,19,14,18)(12,15,20,13,17)$ | |
$ 10, 10 $ | $32$ | $10$ | $( 1, 6, 9,14,18,11,16,19, 4, 8)( 2, 5,10,13,17,12,15,20, 3, 7)$ | |
$ 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: | $320=2^{6} \cdot 5$ | 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: | 320.1636 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 5A1 | 5A2 | 10A1 | 10A3 | ||
Size | 1 | 1 | 5 | 5 | 5 | 5 | 5 | 5 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 32 | 32 | 32 | 32 | |
2 P | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2B | 2B | 2C | 2D | 2D | 2C | 5A2 | 5A1 | 5A1 | 5A2 | |
5 P | 1A | 2A | 2F | 2C | 2D | 2B | 2E | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 1A | 1A | 2A | 2A | |
Type | |||||||||||||||||||||
320.1636.1a | R | ||||||||||||||||||||
320.1636.1b | R | ||||||||||||||||||||
320.1636.1c | R | ||||||||||||||||||||
320.1636.1d | R | ||||||||||||||||||||
320.1636.2a1 | R | ||||||||||||||||||||
320.1636.2a2 | R | ||||||||||||||||||||
320.1636.2b1 | R | ||||||||||||||||||||
320.1636.2b2 | R | ||||||||||||||||||||
320.1636.5a | R | ||||||||||||||||||||
320.1636.5b | R | ||||||||||||||||||||
320.1636.5c | R | ||||||||||||||||||||
320.1636.5d | R | ||||||||||||||||||||
320.1636.5e | R | ||||||||||||||||||||
320.1636.5f | R | ||||||||||||||||||||
320.1636.5g | R | ||||||||||||||||||||
320.1636.5h | R | ||||||||||||||||||||
320.1636.5i | R | ||||||||||||||||||||
320.1636.5j | R | ||||||||||||||||||||
320.1636.5k | R | ||||||||||||||||||||
320.1636.5l | R |
magma: CharacterTable(G);