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Magma
magma: G := TransitiveGroup(25, 31);
Group action invariants
Degree $n$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $31$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_5^2:\OD_{16}$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | yes | 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: | (1,12,19,5,7,18,25,6,13,24)(2,17,20,10,8,23,21,11,14,4)(3,22,16,15,9), (1,21,18,8,22,2,10,20)(3,19,17,24,25,9,6,4)(5,12,16,15,23,11,7,13) | 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, $C_2^2$ $8$: $C_4\times C_2$ $16$: $C_8:C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: None
Low degree siblings
10T28, 20T104, 20T107, 20T109, 20T115, 40T397, 40T398, 40T399, 40T400Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
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 $ | $25$ | $4$ | $( 2, 3, 5, 4)( 6,11,21,16)( 7,13,25,19)( 8,15,24,17)( 9,12,23,20)(10,14,22,18)$ |
$ 4, 4, 4, 4, 4, 4, 1 $ | $25$ | $4$ | $( 2, 4, 5, 3)( 6,16,21,11)( 7,19,25,13)( 8,17,24,15)( 9,20,23,12)(10,18,22,14)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $25$ | $2$ | $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $10$ | $2$ | $( 2, 6)( 3,11)( 4,16)( 5,21)( 8,12)( 9,17)(10,22)(14,18)(15,23)(20,24)$ |
$ 8, 8, 8, 1 $ | $50$ | $8$ | $( 2, 8, 4,17, 5,24, 3,15)( 6,20,16,23,21,12,11, 9)( 7,22,19,14,25,10,13,18)$ |
$ 8, 8, 8, 1 $ | $50$ | $8$ | $( 2, 9, 4,20, 5,23, 3,12)( 6,15,16, 8,21,17,11,24)( 7,18,19,22,25,14,13,10)$ |
$ 4, 4, 4, 4, 4, 4, 1 $ | $50$ | $4$ | $( 2,11, 5,16)( 3,21, 4, 6)( 7,13,25,19)( 8,23,24, 9)(10,18,22,14)(12,15,20,17)$ |
$ 8, 8, 8, 1 $ | $50$ | $8$ | $( 2,12, 3,23, 5,20, 4, 9)( 6,24,11,17,21, 8,16,15)( 7,10,13,14,25,22,19,18)$ |
$ 8, 8, 8, 1 $ | $50$ | $8$ | $( 2,15, 3,24, 5,17, 4, 8)( 6, 9,11,12,21,23,16,20)( 7,18,13,10,25,14,19,22)$ |
$ 5, 5, 5, 5, 5 $ | $16$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ |
$ 10, 10, 5 $ | $40$ | $10$ | $( 1, 2, 7, 8,13,14,19,20,25,21)( 3,12, 9,18,15,24,16, 5,22, 6)( 4,17,10,23,11)$ |
$ 5, 5, 5, 5, 5 $ | $8$ | $5$ | $( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $400=2^{4} \cdot 5^{2}$ | 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: | 400.206 | magma: IdentifyGroup(G);
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Character table: |
2 4 4 4 4 3 3 3 3 3 3 . 1 1 5 2 . . . 1 . . . . . 2 1 2 1a 4a 4b 2a 2b 8a 8b 4c 8c 8d 5a 10a 5b 2P 1a 2a 2a 1a 1a 4b 4b 2a 4a 4a 5a 5b 5b 3P 1a 4b 4a 2a 2b 8d 8c 4c 8b 8a 5a 10a 5b 5P 1a 4a 4b 2a 2b 8a 8b 4c 8c 8d 1a 2b 1a 7P 1a 4b 4a 2a 2b 8d 8c 4c 8b 8a 5a 10a 5b X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 -1 -1 1 -1 1 -1 1 -1 1 X.3 1 1 1 1 -1 1 -1 -1 -1 1 1 -1 1 X.4 1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 X.5 1 -1 -1 1 -1 B B 1 -B -B 1 -1 1 X.6 1 -1 -1 1 -1 -B -B 1 B B 1 -1 1 X.7 1 -1 -1 1 1 B -B -1 B -B 1 1 1 X.8 1 -1 -1 1 1 -B B -1 -B B 1 1 1 X.9 2 A -A -2 . . . . . . 2 . 2 X.10 2 -A A -2 . . . . . . 2 . 2 X.11 8 . . . -4 . . . . . -2 1 3 X.12 8 . . . 4 . . . . . -2 -1 3 X.13 16 . . . . . . . . . 1 . -4 A = -2*E(4) = -2*Sqrt(-1) = -2i B = -E(4) = -Sqrt(-1) = -i |
magma: CharacterTable(G);