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Magma
magma: G := TransitiveGroup(15, 9);
Group action invariants
Degree $n$: | $15$ | 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: | $C_5^2 : C_3$ | ||
CHM label: | $[5^{2}]3$ | ||
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: | (1,13,10,7,4)(2,5,8,11,14), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 5: None
Low degree siblings
15T9, 25T6Siblings 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 $ | $1$ | $1$ | $()$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $3$ | $5$ | $( 2, 5, 8,11,14)( 3,15,12, 9, 6)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $3$ | $5$ | $( 2, 8,14, 5,11)( 3,12, 6,15, 9)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $3$ | $5$ | $( 2,11, 5,14, 8)( 3, 9,15, 6,12)$ |
$ 5, 5, 1, 1, 1, 1, 1 $ | $3$ | $5$ | $( 2,14,11, 8, 5)( 3, 6, 9,12,15)$ |
$ 3, 3, 3, 3, 3 $ | $25$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)$ |
$ 3, 3, 3, 3, 3 $ | $25$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)$ |
$ 5, 5, 5 $ | $3$ | $5$ | $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,12, 6,15, 9)$ |
$ 5, 5, 5 $ | $3$ | $5$ | $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$ |
$ 5, 5, 5 $ | $3$ | $5$ | $( 1, 7,13, 4,10)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$ |
$ 5, 5, 5 $ | $3$ | $5$ | $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,15,12, 9, 6)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $75=3 \cdot 5^{2}$ | 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: | 75.2 | magma: IdentifyGroup(G);
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Character table: |
3 1 . . . . 1 1 . . . . 5 2 2 2 2 2 . . 2 2 2 2 1a 5a 5b 5c 5d 3a 3b 5e 5f 5g 5h 2P 1a 5b 5d 5a 5c 3b 3a 5f 5g 5h 5e 3P 1a 5c 5a 5d 5b 1a 1a 5h 5e 5f 5g 5P 1a 1a 1a 1a 1a 3b 3a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 1 D /D 1 1 1 1 X.3 1 1 1 1 1 /D D 1 1 1 1 X.4 3 A /B B /A . . C *C C *C X.5 3 B A /A /B . . *C C *C C X.6 3 /B /A A B . . *C C *C C X.7 3 /A B /B A . . C *C C *C X.8 3 C *C *C C . . B A /B /A X.9 3 C *C *C C . . /B /A B A X.10 3 *C C C *C . . /A B A /B X.11 3 *C C C *C . . A /B /A B A = 2*E(5)+E(5)^3 B = 2*E(5)^3+E(5)^4 C = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 D = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 |
magma: CharacterTable(G);