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Magma
magma: G := TransitiveGroup(25, 16);
Group action invariants
Degree $n$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_5^2:S_3$ | ||
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: | (1,12,8)(2,6,13)(3,5,18)(4,24,23)(9,25,17)(10,19,22)(11,14,21)(15,20,16), (1,2,3,4,5)(6,21,8,23,10,25,7,22,9,24)(11,20,13,17,15,19,12,16,14,18) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: None
Low degree siblings
15T13, 15T14, 30T37, 30T38Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $( 6,25)( 7,21)( 8,22)( 9,23)(10,24)(11,19)(12,20)(13,16)(14,17)(15,18)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $50$ | $3$ | $( 2, 6,25)( 3,11,19)( 4,16,13)( 5,21, 7)( 8,10,20)( 9,15,14)(12,24,22) (17,18,23)$ |
$ 5, 5, 5, 5, 5 $ | $3$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ |
$ 10, 10, 5 $ | $15$ | $10$ | $( 1, 2, 3, 4, 5)( 6,21, 8,23,10,25, 7,22, 9,24)(11,20,13,17,15,19,12,16,14,18)$ |
$ 10, 10, 5 $ | $15$ | $10$ | $( 1, 2, 7, 8,13,14,19,20,25,21)( 3,12, 9,18,15,24,16, 5,22, 6)( 4,17,10,23,11)$ |
$ 5, 5, 5, 5, 5 $ | $3$ | $5$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$ |
$ 10, 10, 5 $ | $15$ | $10$ | $( 1, 3,13,15,25,22, 7, 9,19,16)( 2, 8,14,20,21)( 4,18,11, 5,23,12,10,24,17, 6)$ |
$ 5, 5, 5, 5, 5 $ | $3$ | $5$ | $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)$ |
$ 10, 10, 5 $ | $15$ | $10$ | $( 1, 4, 2, 5, 3)( 6,23, 7,24, 8,25, 9,21,10,22)(11,17,12,18,13,19,14,20,15,16)$ |
$ 5, 5, 5, 5, 5 $ | $3$ | $5$ | $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)$ |
$ 5, 5, 5, 5, 5 $ | $6$ | $5$ | $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$ |
$ 5, 5, 5, 5, 5 $ | $6$ | $5$ | $( 1, 9,12,20,23)( 2,10,13,16,24)( 3, 6,14,17,25)( 4, 7,15,18,21) ( 5, 8,11,19,22)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $150=2 \cdot 3 \cdot 5^{2}$ | 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: | 150.5 | magma: IdentifyGroup(G);
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Character table: |
2 1 1 . 1 1 1 1 1 1 1 1 . . 3 1 . 1 . . . . . . . . . . 5 2 1 . 2 1 1 2 1 2 1 2 2 2 1a 2a 3a 5a 10a 10b 5b 10c 5c 10d 5d 5e 5f 2P 1a 1a 3a 5b 5b 5d 5d 5c 5a 5a 5c 5f 5e 3P 1a 2a 1a 5c 10d 10a 5a 10b 5d 10c 5b 5f 5e 5P 1a 2a 3a 1a 2a 2a 1a 2a 1a 2a 1a 1a 1a 7P 1a 2a 3a 5b 10b 10c 5d 10d 5a 10a 5c 5f 5e 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 2 . -1 2 . . 2 . 2 . 2 2 2 X.4 3 -1 . A D /E /B /D B E /A F *F X.5 3 -1 . B E D A /E /A /D /B *F F X.6 3 -1 . /A /D E B D /B /E A F *F X.7 3 -1 . /B /E /D /A E A D B *F F X.8 3 1 . A -D -/E /B -/D B -E /A F *F X.9 3 1 . B -E -D A -/E /A -/D /B *F F X.10 3 1 . /A -/D -E B -D /B -/E A F *F X.11 3 1 . /B -/E -/D /A -E A -D B *F F X.12 6 . . C . . *C . *C . C G *G X.13 6 . . *C . . C . C . *C *G G A = 2*E(5)^3+E(5)^4 B = E(5)^2+2*E(5)^4 C = -2*E(5)-2*E(5)^4 = 1-Sqrt(5) = 1-r5 D = -E(5)^4 E = -E(5)^2 F = -E(5)^2-E(5)^3 = (1+Sqrt(5))/2 = 1+b5 G = E(5)+2*E(5)^2+2*E(5)^3+E(5)^4 = (-3-Sqrt(5))/2 = -2-b5 |
magma: CharacterTable(G);