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Magma
magma: G := TransitiveGroup(10, 33);
Group action invariants
Degree $n$: | $10$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $33$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $F_5 \wr C_2$ | ||
CHM label: | $[F(5)^{2}]2$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (2,4,6,8,10), (1,6)(2,7)(3,8)(4,9)(5,10), (2,4,8,6) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $D_{4}$ x 2, $C_4\times C_2$ $16$: $C_2^2:C_4$ $32$: $C_4\wr C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 5: None
Low degree siblings
20T155, 20T161, 20T167, 20T169, 25T50, 40T874, 40T875, 40T876, 40T877, 40T878, 40T879, 40T880, 40T881, 40T882, 40T883Siblings 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 | Index | Representative |
1A | $1^{10}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{2},1^{6}$ | $10$ | $2$ | $2$ | $( 4,10)( 6, 8)$ |
2B | $2^{5}$ | $20$ | $2$ | $5$ | $( 1, 2)( 3,10)( 4, 9)( 5, 8)( 6, 7)$ |
2C | $2^{4},1^{2}$ | $25$ | $2$ | $4$ | $( 2,10)( 3, 9)( 4, 8)( 5, 7)$ |
4A1 | $4,1^{6}$ | $10$ | $4$ | $3$ | $( 2,10, 6, 8)$ |
4A-1 | $4,1^{6}$ | $10$ | $4$ | $3$ | $(1,9,3,5)$ |
4B1 | $4^{2},1^{2}$ | $25$ | $4$ | $6$ | $( 2, 4,10, 8)( 3, 7, 9, 5)$ |
4B-1 | $4^{2},1^{2}$ | $25$ | $4$ | $6$ | $( 2, 8,10, 4)( 3, 5, 9, 7)$ |
4C | $4^{2},1^{2}$ | $50$ | $4$ | $6$ | $( 2, 8, 6,10)( 3, 5, 9, 7)$ |
4D1 | $4,2^{2},1^{2}$ | $50$ | $4$ | $5$ | $(1,7,9,3)(2,8)(4,6)$ |
4D-1 | $4,2^{2},1^{2}$ | $50$ | $4$ | $5$ | $( 1, 7)( 3, 5)( 4, 8,10, 6)$ |
4E | $4^{2},2$ | $100$ | $4$ | $7$ | $( 1, 4, 7, 2)( 3,10, 5, 6)( 8, 9)$ |
5A | $5,1^{5}$ | $8$ | $5$ | $4$ | $( 2, 4, 6, 8,10)$ |
5B | $5^{2}$ | $16$ | $5$ | $8$ | $( 1, 7, 3, 9, 5)( 2,10, 8, 6, 4)$ |
8A1 | $8,2$ | $100$ | $8$ | $8$ | $( 1, 6)( 2, 3, 4, 7,10, 9, 8, 5)$ |
8A-1 | $8,2$ | $100$ | $8$ | $8$ | $( 1, 6)( 2, 7, 8, 3,10, 5, 4, 9)$ |
10A | $5,2^{2},1$ | $40$ | $10$ | $6$ | $( 1, 7, 3, 9, 5)( 2, 4)( 6,10)$ |
10B | $10$ | $80$ | $10$ | $9$ | $( 1, 4, 5,10, 9, 6, 3, 2, 7, 8)$ |
20A1 | $5,4,1$ | $40$ | $20$ | $7$ | $( 1, 7, 3, 9, 5)( 2, 6, 4,10)$ |
20A-1 | $5,4,1$ | $40$ | $20$ | $7$ | $( 1, 7, 5, 9)( 2,10, 8, 6, 4)$ |
Malle's constant $a(G)$: $1/2$
magma: ConjugacyClasses(G);
Group invariants
Order: | $800=2^{5} \cdot 5^{2}$ | 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: | 800.1191 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 4A1 | 4A-1 | 4B1 | 4B-1 | 4C | 4D1 | 4D-1 | 4E | 5A | 5B | 8A1 | 8A-1 | 10A | 10B | 20A1 | 20A-1 | ||
Size | 1 | 10 | 20 | 25 | 10 | 10 | 25 | 25 | 50 | 50 | 50 | 100 | 8 | 16 | 100 | 100 | 40 | 80 | 40 | 40 | |
2 P | 1A | 1A | 1A | 1A | 2A | 2A | 2C | 2C | 2C | 2A | 2A | 2C | 5A | 5B | 4B1 | 4B-1 | 5A | 5B | 10A | 10A | |
5 P | 1A | 2A | 2B | 2C | 4A1 | 4A-1 | 4B1 | 4B-1 | 4C | 4D1 | 4D-1 | 4E | 1A | 1A | 8A1 | 8A-1 | 2A | 2B | 4A1 | 4A-1 | |
Type | |||||||||||||||||||||
800.1191.1a | R | ||||||||||||||||||||
800.1191.1b | R | ||||||||||||||||||||
800.1191.1c | R | ||||||||||||||||||||
800.1191.1d | R | ||||||||||||||||||||
800.1191.1e1 | C | ||||||||||||||||||||
800.1191.1e2 | C | ||||||||||||||||||||
800.1191.1f1 | C | ||||||||||||||||||||
800.1191.1f2 | C | ||||||||||||||||||||
800.1191.2a | R | ||||||||||||||||||||
800.1191.2b | R | ||||||||||||||||||||
800.1191.2c1 | C | ||||||||||||||||||||
800.1191.2c2 | C | ||||||||||||||||||||
800.1191.2d1 | C | ||||||||||||||||||||
800.1191.2d2 | C | ||||||||||||||||||||
800.1191.8a | R | ||||||||||||||||||||
800.1191.8b | R | ||||||||||||||||||||
800.1191.8c1 | C | ||||||||||||||||||||
800.1191.8c2 | C | ||||||||||||||||||||
800.1191.16a | R | ||||||||||||||||||||
800.1191.16b | R |
magma: CharacterTable(G);