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Magma
magma: G := TransitiveGroup(25, 39);
Group action invariants
Degree $n$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_5^2:F_5$ | ||
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: | (1,16,12,21)(2,18,11,24)(3,20,15,22)(4,17,14,25)(5,19,13,23)(6,7,9,8), (1,15,21,8,20)(2,11,22,9,16)(3,12,23,10,17)(4,13,24,6,18)(5,14,25,7,19) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $10$: $D_{5}$ $20$: $F_5$, 20T2 $100$: 20T26 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $F_5$
Low degree siblings
25T36Siblings 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 $ | $10$ | $5$ | $( 6, 8,10, 7, 9)(11,15,14,13,12)(16,17,18,19,20)(21,24,22,25,23)$ | |
$ 4, 4, 4, 4, 4, 4, 1 $ | $125$ | $4$ | $( 2, 3, 5, 4)( 6,16,21,11)( 7,18,25,14)( 8,20,24,12)( 9,17,23,15)(10,19,22,13)$ | |
$ 4, 4, 4, 4, 4, 4, 1 $ | $125$ | $4$ | $( 2, 4, 5, 3)( 6,11,21,16)( 7,14,25,18)( 8,12,24,20)( 9,15,23,17)(10,13,22,19)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $25$ | $2$ | $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$ | |
$ 10, 10, 2, 2, 1 $ | $50$ | $10$ | $( 2, 5)( 3, 4)( 6,22, 9,24, 7,21,10,23, 8,25)(11,18,12,17,13,16,14,20,15,19)$ | |
$ 10, 10, 2, 2, 1 $ | $50$ | $10$ | $( 2, 5)( 3, 4)( 6,23, 7,22, 8,21, 9,25,10,24)(11,20,13,18,15,16,12,19,14,17)$ | |
$ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ | |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,11,19,23)( 2, 7,12,20,24)( 3, 8,13,16,25)( 4, 9,14,17,21) ( 5,10,15,18,22)$ | |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,12,17,24)( 2, 7,13,18,25)( 3, 8,14,19,21)( 4, 9,15,20,22) ( 5,10,11,16,23)$ | |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,13,20,25)( 2, 7,14,16,21)( 3, 8,15,17,22)( 4, 9,11,18,23) ( 5,10,12,19,24)$ | |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,14,18,21)( 2, 7,15,19,22)( 3, 8,11,20,23)( 4, 9,12,16,24) ( 5,10,13,17,25)$ | |
$ 5, 5, 5, 5, 5 $ | $20$ | $5$ | $( 1, 6,15,16,22)( 2, 7,11,17,23)( 3, 8,12,18,24)( 4, 9,13,19,25) ( 5,10,14,20,21)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $500=2^{2} \cdot 5^{3}$ | 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: | 500.21 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 4A1 | 4A-1 | 5A | 5B1 | 5B2 | 5C | 5D1 | 5D-1 | 5D2 | 5D-2 | 10A1 | 10A3 | ||
Size | 1 | 25 | 125 | 125 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 50 | 50 | |
2 P | 1A | 1A | 2A | 2A | 5A | 5B2 | 5B1 | 5D2 | 5C | 5D-2 | 5D1 | 5D-1 | 5B1 | 5B2 | |
5 P | 1A | 2A | 4A1 | 4A-1 | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2A | 2A | |
Type | |||||||||||||||
500.21.1a | R | ||||||||||||||
500.21.1b | R | ||||||||||||||
500.21.1c1 | C | ||||||||||||||
500.21.1c2 | C | ||||||||||||||
500.21.2a1 | R | ||||||||||||||
500.21.2a2 | R | ||||||||||||||
500.21.2b1 | S | ||||||||||||||
500.21.2b2 | S | ||||||||||||||
500.21.4a | R | ||||||||||||||
500.21.4b1 | C | ||||||||||||||
500.21.4b2 | C | ||||||||||||||
500.21.4b3 | C | ||||||||||||||
500.21.4b4 | C | ||||||||||||||
500.21.20a | R |
magma: CharacterTable(G);