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Magma
magma: G := TransitiveGroup(16, 24);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2^2 : C_8$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $8$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,3,5,7,10,11,14,15)(2,4,6,8,9,12,13,16), (1,10)(2,9)(3,4)(5,14)(6,13)(7,8)(11,12)(15,16) | 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, $C_2^2$ $8$: $D_{4}$ x 2, $C_8$ x 2, $C_4\times C_2$ $16$: $C_8:C_2$, $C_2^2:C_4$, $C_8\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 8: $C_8$, $C_8:C_2$, $C_2^2:C_4$
Low degree siblings
16T24, 32T10Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 3,12)( 4,11)( 7,16)( 8,15)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 2)( 3,11)( 4,12)( 5, 6)( 7,15)( 8,16)( 9,10)(13,14)$ |
$ 8, 8 $ | $2$ | $8$ | $( 1, 3, 5, 7,10,11,14,15)( 2, 4, 6, 8, 9,12,13,16)$ |
$ 8, 8 $ | $2$ | $8$ | $( 1, 3,13,16,10,11, 6, 8)( 2, 4,14,15, 9,12, 5, 7)$ |
$ 8, 8 $ | $2$ | $8$ | $( 1, 4, 5, 8,10,12,14,16)( 2, 3, 6, 7, 9,11,13,15)$ |
$ 8, 8 $ | $2$ | $8$ | $( 1, 4,13,15,10,12, 6, 7)( 2, 3,14,16, 9,11, 5, 8)$ |
$ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 5,10,14)( 2, 6, 9,13)( 3, 7,11,15)( 4, 8,12,16)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 5,10,14)( 2, 6, 9,13)( 3,16,11, 8)( 4,15,12, 7)$ |
$ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 6,10,13)( 2, 5, 9,14)( 3, 8,11,16)( 4, 7,12,15)$ |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 6,10,13)( 2, 5, 9,14)( 3,15,11, 7)( 4,16,12, 8)$ |
$ 8, 8 $ | $2$ | $8$ | $( 1, 7, 6,12,10,15,13, 4)( 2, 8, 5,11, 9,16,14, 3)$ |
$ 8, 8 $ | $2$ | $8$ | $( 1, 7,14, 3,10,15, 5,11)( 2, 8,13, 4, 9,16, 6,12)$ |
$ 8, 8 $ | $2$ | $8$ | $( 1, 8, 6,11,10,16,13, 3)( 2, 7, 5,12, 9,15,14, 4)$ |
$ 8, 8 $ | $2$ | $8$ | $( 1, 8,14, 4,10,16, 5,12)( 2, 7,13, 3, 9,15, 6,11)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,13)( 6,14)( 7,16)( 8,15)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,14)( 6,13)( 7,15)( 8,16)$ |
$ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,13,10, 6)( 2,14, 9, 5)( 3,16,11, 8)( 4,15,12, 7)$ |
$ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,14,10, 5)( 2,13, 9, 6)( 3,15,11, 7)( 4,16,12, 8)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $32=2^{5}$ | 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: | $2$ | ||
Label: | 32.5 | magma: IdentifyGroup(G);
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Character table: |
2 5 4 5 4 4 4 4 4 5 4 5 4 4 4 4 4 5 5 5 5 1a 2a 2b 2c 8a 8b 8c 8d 4a 4b 4c 4d 8e 8f 8g 8h 2d 2e 4e 4f 2P 1a 1a 1a 1a 4a 4e 4a 4e 2e 2e 2e 2e 4c 4f 4c 4f 1a 1a 2e 2e 3P 1a 2a 2b 2c 8f 8e 8h 8g 4f 4d 4e 4b 8b 8a 8d 8c 2d 2e 4c 4a 5P 1a 2a 2b 2c 8c 8d 8a 8b 4a 4b 4c 4d 8g 8h 8e 8f 2d 2e 4e 4f 7P 1a 2a 2b 2c 8h 8g 8f 8e 4f 4d 4e 4b 8d 8c 8b 8a 2d 2e 4c 4a X.1 1 1 1 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 1 1 1 1 X.3 1 -1 1 -1 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 1 1 1 1 X.5 1 -1 -1 1 A -A -A A -B B B -B /A -/A -/A /A 1 -1 -B B X.6 1 -1 -1 1 -/A /A /A -/A B -B -B B -A A A -A 1 -1 B -B X.7 1 -1 -1 1 /A -/A -/A /A B -B -B B A -A -A A 1 -1 B -B X.8 1 -1 -1 1 -A A A -A -B B B -B -/A /A /A -/A 1 -1 -B B X.9 1 -1 1 -1 B -B B -B -1 1 -1 1 B -B B -B 1 1 -1 -1 X.10 1 -1 1 -1 -B B -B B -1 1 -1 1 -B B -B B 1 1 -1 -1 X.11 1 1 -1 -1 A A -A -A -B -B B B -/A -/A /A /A 1 -1 -B B X.12 1 1 -1 -1 -/A -/A /A /A B B -B -B A A -A -A 1 -1 B -B X.13 1 1 -1 -1 /A /A -/A -/A B B -B -B -A -A A A 1 -1 B -B X.14 1 1 -1 -1 -A -A A A -B -B B B /A /A -/A -/A 1 -1 -B B X.15 1 1 1 1 B B B B -1 -1 -1 -1 -B -B -B -B 1 1 -1 -1 X.16 1 1 1 1 -B -B -B -B -1 -1 -1 -1 B B B B 1 1 -1 -1 X.17 2 . -2 . . . . . -2 . 2 . . . . . -2 2 2 -2 X.18 2 . -2 . . . . . 2 . -2 . . . . . -2 2 -2 2 X.19 2 . 2 . . . . . C . C . . . . . -2 -2 -C -C X.20 2 . 2 . . . . . -C . -C . . . . . -2 -2 C C A = -E(8) B = -E(4) = -Sqrt(-1) = -i C = -2*E(4) = -2*Sqrt(-1) = -2i |
magma: CharacterTable(G);