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Magma
magma: G := TransitiveGroup(15, 27);
Group action invariants
Degree $n$: | $15$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_5^2:(C_4\times S_3)$ | ||
CHM label: | $[5^{2}:4]S(3)$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,13,10,7,4)(2,5,8,11,14), (1,11)(2,7)(4,14)(5,10)(8,13), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15), (1,7,4,13)(2,14,8,11)(3,6,12,9) | 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, $C_2^2$ $6$: $S_3$ $8$: $C_4\times C_2$ $12$: $D_{6}$ $24$: $S_3 \times C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 5: None
Low degree siblings
15T27, 25T43, 30T150 x 2, 30T153 x 2, 30T155 x 2Siblings 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 $ | $12$ | $5$ | $( 1,13,10, 7, 4)( 2, 5, 8,11,14)$ |
$ 5, 5, 5 $ | $12$ | $5$ | $( 1,13,10, 7, 4)( 2, 8,14, 5,11)( 3,15,12, 9, 6)$ |
$ 3, 3, 3, 3, 3 $ | $50$ | $3$ | $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$ |
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $( 2,12)( 3, 8)( 5,15)( 6,11)( 9,14)$ |
$ 10, 5 $ | $60$ | $10$ | $( 1,13,10, 7, 4)( 2,12, 5,15, 8, 3,11, 6,14, 9)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $25$ | $2$ | $( 3, 6)( 4,13)( 5,14)( 7,10)( 8,11)( 9,15)$ |
$ 6, 6, 3 $ | $50$ | $6$ | $( 1,11, 3)( 2,12, 7, 5, 9,10)( 4, 8, 6,13,14,15)$ |
$ 2, 2, 2, 2, 2, 2, 2, 1 $ | $15$ | $2$ | $( 2,12)( 3,11)( 4,13)( 5, 9)( 6, 8)( 7,10)(14,15)$ |
$ 10, 2, 2, 1 $ | $60$ | $10$ | $( 1,13)( 2,12, 5, 9, 8, 6,11, 3,14,15)( 4,10)$ |
$ 4, 4, 4, 1, 1, 1 $ | $25$ | $4$ | $( 3, 9, 6,15)( 4, 7,13,10)( 5, 8,14,11)$ |
$ 12, 3 $ | $50$ | $12$ | $( 1,11,15,13, 5, 3, 4, 2,12, 7, 8, 9)( 6,10,14)$ |
$ 4, 4, 4, 2, 1 $ | $75$ | $4$ | $( 2,12)( 3,14, 6, 5)( 4, 7,13,10)( 8, 9,11,15)$ |
$ 4, 4, 4, 1, 1, 1 $ | $25$ | $4$ | $( 3,15, 6, 9)( 4,10,13, 7)( 5,11,14, 8)$ |
$ 12, 3 $ | $50$ | $12$ | $( 1,11, 9,13, 2,12, 7,14, 3,10, 8,15)( 4, 5, 6)$ |
$ 4, 4, 4, 2, 1 $ | $75$ | $4$ | $( 2,12)( 3, 5, 6,14)( 4,10,13, 7)( 8,15,11, 9)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $600=2^{3} \cdot 3 \cdot 5^{2}$ | 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: | 600.151 | magma: IdentifyGroup(G);
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Character table: |
2 3 1 1 2 3 1 3 2 3 1 3 2 3 3 2 3 3 1 . . 1 . . 1 1 . . 1 1 . 1 1 . 5 2 2 2 . 1 1 . . 1 1 . . . . . . 1a 5a 5b 3a 2a 10a 2b 6a 2c 10b 4a 12a 4b 4c 12b 4d 2P 1a 5a 5b 3a 1a 5b 1a 3a 1a 5a 2b 6a 2b 2b 6a 2b 3P 1a 5a 5b 1a 2a 10a 2b 2b 2c 10b 4c 4c 4d 4a 4a 4b 5P 1a 1a 1a 3a 2a 2a 2b 6a 2c 2c 4a 12a 4b 4c 12b 4d 7P 1a 5a 5b 3a 2a 10a 2b 6a 2c 10b 4c 12b 4d 4a 12a 4b 11P 1a 5a 5b 3a 2a 10a 2b 6a 2c 10b 4c 12b 4d 4a 12a 4b X.1 1 1 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 -1 -1 1 X.3 1 1 1 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 -1 -1 -1 X.5 1 1 1 1 -1 -1 -1 -1 1 1 A A -A -A -A A X.6 1 1 1 1 -1 -1 -1 -1 1 1 -A -A A A A -A X.7 1 1 1 1 1 1 -1 -1 -1 -1 A A A -A -A -A X.8 1 1 1 1 1 1 -1 -1 -1 -1 -A -A -A A A A X.9 2 2 2 -1 . . 2 -1 . . -2 1 . -2 1 . X.10 2 2 2 -1 . . 2 -1 . . 2 -1 . 2 -1 . X.11 2 2 2 -1 . . -2 1 . . B -A . -B A . X.12 2 2 2 -1 . . -2 1 . . -B A . B -A . X.13 12 -3 2 . . . . . -4 1 . . . . . . X.14 12 -3 2 . . . . . 4 -1 . . . . . . X.15 12 2 -3 . -4 1 . . . . . . . . . . X.16 12 2 -3 . 4 -1 . . . . . . . . . . A = -E(4) = -Sqrt(-1) = -i B = -2*E(4) = -2*Sqrt(-1) = -2i |
magma: CharacterTable(G);