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Magma
magma: G := TransitiveGroup(32, 31);
Group action invariants
Degree $n$: | $32$ | 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: | $D_{16}$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $32$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,6)(2,5)(3,7)(4,8)(9,29)(10,30)(11,31)(12,32)(13,26)(14,25)(15,28)(16,27)(17,22)(18,21)(19,23)(20,24), (1,11)(2,12)(3,10)(4,9)(5,7)(6,8)(13,30)(14,29)(15,32)(16,31)(17,25)(18,26)(19,28)(20,27)(21,22)(23,24) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $D_{4}$ $16$: $D_{8}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Low degree siblings
16T56 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, 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, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 2)( 3, 4)( 5,29)( 6,30)( 7,31)( 8,32)( 9,28)(10,27)(11,26)(12,25)(13,24) (14,23)(15,22)(16,21)(17,20)(18,19)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)$ |
$ 16, 16 $ | $2$ | $16$ | $( 1, 5, 9,15,17,24,26,31, 3, 8,12,14,19,21,27,30)( 2, 6,10,16,18,23,25,32, 4, 7,11,13,20,22,28,29)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,29)(10,30)(11,31)(12,32)(13,26)(14,25)(15,28) (16,27)(17,22)(18,21)(19,23)(20,24)$ |
$ 16, 16 $ | $2$ | $16$ | $( 1, 8, 9,14,17,21,26,30, 3, 5,12,15,19,24,27,31)( 2, 7,10,13,18,22,25,29, 4, 6,11,16,20,23,28,32)$ |
$ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1, 9,17,26, 3,12,19,27)( 2,10,18,25, 4,11,20,28)( 5,15,24,31, 8,14,21,30) ( 6,16,23,32, 7,13,22,29)$ |
$ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,12,17,27, 3, 9,19,26)( 2,11,18,28, 4,10,20,25)( 5,14,24,30, 8,15,21,31) ( 6,13,23,29, 7,16,22,32)$ |
$ 16, 16 $ | $2$ | $16$ | $( 1,14,26, 5,19,31, 9,21, 3,15,27, 8,17,30,12,24)( 2,13,25, 6,20,32,10,22, 4, 16,28, 7,18,29,11,23)$ |
$ 16, 16 $ | $2$ | $16$ | $( 1,15,26, 8,19,30, 9,24, 3,14,27, 5,17,31,12,21)( 2,16,25, 7,20,29,10,23, 4, 13,28, 6,18,32,11,22)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,17, 3,19)( 2,18, 4,20)( 5,24, 8,21)( 6,23, 7,22)( 9,26,12,27)(10,25,11,28) (13,29,16,32)(14,30,15,31)$ |
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: | $4$ | ||
Label: | 32.18 | magma: IdentifyGroup(G);
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Character table: |
2 5 2 5 4 2 4 4 4 4 4 4 1a 2a 2b 16a 2c 16b 8a 8b 16c 16d 4a 2P 1a 1a 1a 8a 1a 8a 4a 4a 8b 8b 2b 3P 1a 2a 2b 16d 2c 16c 8b 8a 16a 16b 4a 5P 1a 2a 2b 16c 2c 16d 8b 8a 16b 16a 4a 7P 1a 2a 2b 16b 2c 16a 8a 8b 16d 16c 4a 11P 1a 2a 2b 16c 2c 16d 8b 8a 16b 16a 4a 13P 1a 2a 2b 16d 2c 16c 8b 8a 16a 16b 4a X.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 X.3 1 -1 1 1 -1 1 1 1 1 1 1 X.4 1 1 1 -1 -1 -1 1 1 -1 -1 1 X.5 2 . 2 . . . -2 -2 . . 2 X.6 2 . 2 A . A . . -A -A -2 X.7 2 . 2 -A . -A . . A A -2 X.8 2 . -2 B . -B -A A -C C . X.9 2 . -2 C . -C A -A B -B . X.10 2 . -2 -C . C A -A -B B . X.11 2 . -2 -B . B -A A C -C . A = -E(8)+E(8)^3 = -Sqrt(2) = -r2 B = -E(16)+E(16)^7 C = -E(16)^3+E(16)^5 |
magma: CharacterTable(G);