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Magma
magma: G := TransitiveGroup(25, 18);
Group action invariants
Degree $n$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $D_5\times F_5$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,2)(3,5)(6,17,21,12)(7,16,22,11)(8,20,23,15)(9,19,24,14)(10,18,25,13), (1,3)(4,5)(6,23)(7,22)(8,21)(9,25)(10,24)(11,18)(12,17)(13,16)(14,20)(15,19), (1,7,13,19,25)(2,8,14,20,21)(3,9,15,16,22)(4,10,11,17,23)(5,6,12,18,24) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $C_4\times C_2$ $10$: $D_{5}$ $20$: $F_5$, $D_{10}$ $40$: $F_{5}\times C_2$, 20T6 Resolvents shown for degrees $\leq 47$
Subfields
Low degree siblings
20T51, 40T168Siblings 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, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
$ 4, 4, 4, 4, 4, 1, 1, 1, 1, 1 $ | $5$ | $4$ | $( 6,11,21,16)( 7,12,22,17)( 8,13,23,18)( 9,14,24,19)(10,15,25,20)$ | |
$ 4, 4, 4, 4, 4, 1, 1, 1, 1, 1 $ | $5$ | $4$ | $( 6,16,21,11)( 7,17,22,12)( 8,18,23,13)( 9,19,24,14)(10,20,25,15)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)(15,20)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)$ | |
$ 4, 4, 4, 4, 4, 2, 2, 1 $ | $25$ | $4$ | $( 2, 5)( 3, 4)( 6,11,21,16)( 7,15,22,20)( 8,14,23,19)( 9,13,24,18) (10,12,25,17)$ | |
$ 4, 4, 4, 4, 4, 2, 2, 1 $ | $25$ | $4$ | $( 2, 5)( 3, 4)( 6,16,21,11)( 7,20,22,15)( 8,19,23,14)( 9,18,24,13) (10,17,25,12)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $25$ | $2$ | $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$ | |
$ 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ | |
$ 20, 5 $ | $10$ | $20$ | $( 1, 2, 3, 4, 5)( 6,12,23,19,10,11,22,18, 9,15,21,17, 8,14,25,16, 7,13,24,20)$ | |
$ 20, 5 $ | $10$ | $20$ | $( 1, 2, 3, 4, 5)( 6,17,23,14,10,16,22,13, 9,20,21,12, 8,19,25,11, 7,18,24,15)$ | |
$ 10, 10, 5 $ | $10$ | $10$ | $( 1, 2, 3, 4, 5)( 6,22, 8,24,10,21, 7,23, 9,25)(11,17,13,19,15,16,12,18,14,20)$ | |
$ 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$ | |
$ 20, 5 $ | $10$ | $20$ | $( 1, 3, 5, 2, 4)( 6,13,25,17, 9,11,23,20, 7,14,21,18,10,12,24,16, 8,15,22,19)$ | |
$ 20, 5 $ | $10$ | $20$ | $( 1, 3, 5, 2, 4)( 6,18,25,12, 9,16,23,15, 7,19,21,13,10,17,24,11, 8,20,22,14)$ | |
$ 10, 10, 5 $ | $10$ | $10$ | $( 1, 3, 5, 2, 4)( 6,23,10,22, 9,21, 8,25, 7,24)(11,18,15,17,14,16,13,20,12,19)$ | |
$ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 6,11,16,21)( 2, 7,12,17,22)( 3, 8,13,18,23)( 4, 9,14,19,24) ( 5,10,15,20,25)$ | |
$ 10, 10, 5 $ | $20$ | $10$ | $( 1, 6,11,16,21)( 2,10,12,20,22, 5, 7,15,17,25)( 3, 9,13,19,23, 4, 8,14,18,24)$ | |
$ 5, 5, 5, 5, 5 $ | $8$ | $5$ | $( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)$ | |
$ 5, 5, 5, 5, 5 $ | $8$ | $5$ | $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $200=2^{3} \cdot 5^{2}$ | 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: | 200.41 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 4A1 | 4A-1 | 4B1 | 4B-1 | 5A1 | 5A2 | 5B | 5C1 | 5C2 | 10A1 | 10A3 | 10B | 20A1 | 20A-1 | 20A3 | 20A-3 | ||
Size | 1 | 5 | 5 | 25 | 5 | 5 | 25 | 25 | 2 | 2 | 4 | 8 | 8 | 10 | 10 | 20 | 10 | 10 | 10 | 10 | |
2 P | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | 5A2 | 5A1 | 5B | 5C2 | 5C1 | 5A1 | 5A2 | 5B | 10A1 | 10A1 | 10A3 | 10A3 | |
5 P | 1A | 2A | 2B | 2C | 4A1 | 4A-1 | 4B1 | 4B-1 | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 2B | 4A1 | 4A-1 | 4A-1 | 4A1 | |
Type | |||||||||||||||||||||
200.41.1a | R | ||||||||||||||||||||
200.41.1b | R | ||||||||||||||||||||
200.41.1c | R | ||||||||||||||||||||
200.41.1d | R | ||||||||||||||||||||
200.41.1e1 | C | ||||||||||||||||||||
200.41.1e2 | C | ||||||||||||||||||||
200.41.1f1 | C | ||||||||||||||||||||
200.41.1f2 | C | ||||||||||||||||||||
200.41.2a1 | R | ||||||||||||||||||||
200.41.2a2 | R | ||||||||||||||||||||
200.41.2b1 | R | ||||||||||||||||||||
200.41.2b2 | R | ||||||||||||||||||||
200.41.2c1 | C | ||||||||||||||||||||
200.41.2c2 | C | ||||||||||||||||||||
200.41.2c3 | C | ||||||||||||||||||||
200.41.2c4 | C | ||||||||||||||||||||
200.41.4a | R | ||||||||||||||||||||
200.41.4b | R | ||||||||||||||||||||
200.41.8a1 | R | ||||||||||||||||||||
200.41.8a2 | R |
magma: CharacterTable(G);