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Magma
magma: G := TransitiveGroup(25, 9);
Group action invariants
Degree $n$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $9$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_5:F_5$ | ||
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: | (1,14,24,6,17)(2,15,25,7,18)(3,11,21,8,19)(4,12,22,9,20)(5,13,23,10,16), (1,25,19,12,10)(2,21,20,13,6)(3,22,16,14,7)(4,23,17,15,8)(5,24,18,11,9), (1,19,25,10)(2,16,24,8)(3,18,23,6)(4,20,22,9)(5,17,21,7)(11,15,13,14) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $20$: $F_5$ x 6 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $F_5$ x 6
Low degree siblings
There are no siblings 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, 4, 1 $ | $25$ | $4$ | $( 2, 3, 5, 4)( 6,14,24,17)( 7,11,23,20)( 8,13,22,18)( 9,15,21,16)(10,12,25,19)$ | |
$ 4, 4, 4, 4, 4, 4, 1 $ | $25$ | $4$ | $( 2, 4, 5, 3)( 6,17,24,14)( 7,20,23,11)( 8,18,22,13)( 9,16,21,15)(10,19,25,12)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $25$ | $2$ | $( 2, 5)( 3, 4)( 6,24)( 7,23)( 8,22)( 9,21)(10,25)(11,20)(12,19)(13,18)(14,17) (15,16)$ | |
$ 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 $ | $4$ | $5$ | $( 1, 6,14,17,24)( 2, 7,15,18,25)( 3, 8,11,19,21)( 4, 9,12,20,22) ( 5,10,13,16,23)$ | |
$ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 7,11,20,23)( 2, 8,12,16,24)( 3, 9,13,17,25)( 4,10,14,18,21) ( 5, 6,15,19,22)$ | |
$ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 8,13,18,22)( 2, 9,14,19,23)( 3,10,15,20,24)( 4, 6,11,16,25) ( 5, 7,12,17,21)$ | |
$ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 9,15,16,21)( 2,10,11,17,22)( 3, 6,12,18,23)( 4, 7,13,19,24) ( 5, 8,14,20,25)$ | |
$ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1,10,12,19,25)( 2, 6,13,20,21)( 3, 7,14,16,22)( 4, 8,15,17,23) ( 5, 9,11,18,24)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $100=2^{2} \cdot 5^{2}$ | 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: | 100.11 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 4A1 | 4A-1 | 5A | 5B | 5C | 5D | 5E | 5F | ||
Size | 1 | 25 | 25 | 25 | 4 | 4 | 4 | 4 | 4 | 4 | |
2 P | 1A | 1A | 2A | 2A | 5D | 5E | 5C | 5A | 5B | 5F | |
5 P | 1A | 2A | 4A1 | 4A-1 | 1A | 1A | 1A | 1A | 1A | 1A | |
Type | |||||||||||
100.11.1a | R | ||||||||||
100.11.1b | R | ||||||||||
100.11.1c1 | C | ||||||||||
100.11.1c2 | C | ||||||||||
100.11.4a | R | ||||||||||
100.11.4b | R | ||||||||||
100.11.4c | R | ||||||||||
100.11.4d | R | ||||||||||
100.11.4e | R | ||||||||||
100.11.4f | R |
magma: CharacterTable(G);