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Magma
magma: G := TransitiveGroup(25, 22);
Group action invariants
Degree $n$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $22$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_5^2:D_5$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $5$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,8)(2,9)(3,10)(4,6)(5,7)(11,22)(12,23)(13,24)(14,25)(15,21), (1,19)(2,20)(3,16)(4,17)(5,18)(6,11)(7,12)(8,13)(9,14)(10,15), (1,3,5,2,4)(6,21,10,25,9,24,8,23,7,22)(11,18,15,17,14,16,13,20,12,19) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $10$: $D_{5}$ x 6 $50$: 25T5 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $D_{5}$
Low degree siblings
25T22 x 5Siblings 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 $ | $1$ | $1$ | $()$ | |
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ | $10$ | $5$ | $( 6, 7, 8, 9,10)(11,13,15,12,14)(16,19,17,20,18)(21,25,24,23,22)$ | |
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ | $10$ | $5$ | $( 6, 8,10, 7, 9)(11,15,14,13,12)(16,17,18,19,20)(21,24,22,25,23)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $25$ | $2$ | $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,20)(12,16)(13,17)(14,18)(15,19)$ | |
$ 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ | |
$ 10, 10, 5 $ | $25$ | $10$ | $( 1, 2, 3, 4, 5)( 6,21, 8,23,10,25, 7,22, 9,24)(11,19,13,16,15,18,12,20,14,17)$ | |
$ 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$ | |
$ 10, 10, 5 $ | $25$ | $10$ | $( 1, 3, 5, 2, 4)( 6,21,10,25, 9,24, 8,23, 7,22)(11,18,15,17,14,16,13,20,12,19)$ | |
$ 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)$ | |
$ 10, 10, 5 $ | $25$ | $10$ | $( 1, 4, 2, 5, 3)( 6,21, 7,22, 8,23, 9,24,10,25)(11,17,12,18,13,19,14,20,15,16)$ | |
$ 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)$ | |
$ 10, 10, 5 $ | $25$ | $10$ | $( 1, 5, 4, 3, 2)( 6,21, 9,24, 7,22,10,25, 8,23)(11,16,14,19,12,17,15,20,13,18)$ | |
$ 5, 5, 5, 5, 5 $ | $10$ | $5$ | $( 1, 6,11,19,23)( 2, 7,12,20,24)( 3, 8,13,16,25)( 4, 9,14,17,21) ( 5,10,15,18,22)$ | |
$ 5, 5, 5, 5, 5 $ | $10$ | $5$ | $( 1, 6,12,17,24)( 2, 7,13,18,25)( 3, 8,14,19,21)( 4, 9,15,20,22) ( 5,10,11,16,23)$ | |
$ 5, 5, 5, 5, 5 $ | $10$ | $5$ | $( 1, 6,13,20,25)( 2, 7,14,16,21)( 3, 8,15,17,22)( 4, 9,11,18,23) ( 5,10,12,19,24)$ | |
$ 5, 5, 5, 5, 5 $ | $10$ | $5$ | $( 1, 6,14,18,21)( 2, 7,15,19,22)( 3, 8,11,20,23)( 4, 9,12,16,24) ( 5,10,13,17,25)$ | |
$ 5, 5, 5, 5, 5 $ | $10$ | $5$ | $( 1, 6,15,16,22)( 2, 7,11,17,23)( 3, 8,12,18,24)( 4, 9,13,19,25) ( 5,10,14,20,21)$ | |
$ 5, 5, 5, 5, 5 $ | $10$ | $5$ | $( 1,11,22, 8,18)( 2,12,23, 9,19)( 3,13,24,10,20)( 4,14,25, 6,16) ( 5,15,21, 7,17)$ | |
$ 5, 5, 5, 5, 5 $ | $10$ | $5$ | $( 1,11,24, 9,20)( 2,12,25,10,16)( 3,13,21, 6,17)( 4,14,22, 7,18) ( 5,15,23, 8,19)$ | |
$ 5, 5, 5, 5, 5 $ | $10$ | $5$ | $( 1,11,21,10,17)( 2,12,22, 6,18)( 3,13,23, 7,19)( 4,14,24, 8,20) ( 5,15,25, 9,16)$ | |
$ 5, 5, 5, 5, 5 $ | $10$ | $5$ | $( 1,11,23, 6,19)( 2,12,24, 7,20)( 3,13,25, 8,16)( 4,14,21, 9,17) ( 5,15,22,10,18)$ | |
$ 5, 5, 5, 5, 5 $ | $10$ | $5$ | $( 1,11,25, 7,16)( 2,12,21, 8,17)( 3,13,22, 9,18)( 4,14,23,10,19) ( 5,15,24, 6,20)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $250=2 \cdot 5^{3}$ | 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: | 250.8 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 5A1 | 5A-1 | 5A2 | 5A-2 | 5B1 | 5B2 | 5C1 | 5C2 | 5D1 | 5D2 | 5E1 | 5E2 | 5F1 | 5F2 | 5G1 | 5G2 | 10A1 | 10A-1 | 10A3 | 10A-3 | ||
Size | 1 | 25 | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 25 | 25 | 25 | 25 | |
2 P | 1A | 1A | 5A2 | 5A-2 | 5A-1 | 5A1 | 5G1 | 5D1 | 5E2 | 5C2 | 5B2 | 5E1 | 5F2 | 5D2 | 5G2 | 5C1 | 5F1 | 5B1 | 5A-2 | 5A2 | 5A1 | 5A-1 | |
5 P | 1A | 2A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | |
Type | |||||||||||||||||||||||
250.8.1a | R | ||||||||||||||||||||||
250.8.1b | R | ||||||||||||||||||||||
250.8.2a1 | R | ||||||||||||||||||||||
250.8.2a2 | R | ||||||||||||||||||||||
250.8.2b1 | R | ||||||||||||||||||||||
250.8.2b2 | R | ||||||||||||||||||||||
250.8.2c1 | R | ||||||||||||||||||||||
250.8.2c2 | R | ||||||||||||||||||||||
250.8.2d1 | R | ||||||||||||||||||||||
250.8.2d2 | R | ||||||||||||||||||||||
250.8.2e1 | R | ||||||||||||||||||||||
250.8.2e2 | R | ||||||||||||||||||||||
250.8.2f1 | R | ||||||||||||||||||||||
250.8.2f2 | R | ||||||||||||||||||||||
250.8.5a1 | C | ||||||||||||||||||||||
250.8.5a2 | C | ||||||||||||||||||||||
250.8.5a3 | C | ||||||||||||||||||||||
250.8.5a4 | C | ||||||||||||||||||||||
250.8.5b1 | C | ||||||||||||||||||||||
250.8.5b2 | C | ||||||||||||||||||||||
250.8.5b3 | C | ||||||||||||||||||||||
250.8.5b4 | C |
magma: CharacterTable(G);