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Magma
magma: G := TransitiveGroup(32, 22);
Group action invariants
Degree $n$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $22$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $\OD_{16}:C_2$ | ||
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: | (1,17)(2,18)(3,20)(4,19)(5,21)(6,22)(7,24)(8,23)(9,11)(10,12)(13,16)(14,15)(25,28)(26,27)(29,31)(30,32), (1,7,11,31,20,22,25,14)(2,8,12,32,19,21,26,13)(3,5,9,30,17,23,28,16)(4,6,10,29,18,24,27,15) | 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, $C_4\times C_2$ $16$: $C_2^2:C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 4
Degree 8: $C_4\times C_2$, $D_4$ x 2, $C_2^2:C_4$ x 2, $(C_8:C_2):C_2$ x 2
Degree 16: $C_2^2 : C_4$, 16T36, 16T41 x 2
Low degree siblings
8T16 x 2, 16T36, 16T41 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 $ | $2$ | $2$ | $( 1, 2)( 3, 4)( 5,24)( 6,23)( 7,21)( 8,22)( 9,10)(11,12)(13,31)(14,32)(15,30) (16,29)(17,18)(19,20)(25,26)(27,28)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,25)(10,26)(11,28)(12,27)(13,30)(14,29)(15,31) (16,32)(17,20)(18,19)(21,23)(22,24)$ |
$ 8, 8, 8, 8 $ | $4$ | $8$ | $( 1, 5,12,15,20,23,26,29)( 2, 6,11,16,19,24,25,30)( 3, 7,10,13,17,22,27,32) ( 4, 8, 9,14,18,21,28,31)$ |
$ 8, 8, 8, 8 $ | $4$ | $8$ | $( 1, 7,11,31,20,22,25,14)( 2, 8,12,32,19,21,26,13)( 3, 5, 9,30,17,23,28,16) ( 4, 6,10,29,18,24,27,15)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,32)( 6,31)( 7,29)( 8,30)(13,23)(14,24)(15,22) (16,21)(17,25)(18,26)(19,27)(20,28)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,11,20,25)( 2,12,19,26)( 3, 9,17,28)( 4,10,18,27)( 5,30,23,16)( 6,29,24,15) ( 7,31,22,14)( 8,32,21,13)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,12,20,26)( 2,11,19,25)( 3,10,17,27)( 4, 9,18,28)( 5,15,23,29)( 6,16,24,30) ( 7,13,22,32)( 8,14,21,31)$ |
$ 8, 8, 8, 8 $ | $4$ | $8$ | $( 1,13,11, 8,20,32,25,21)( 2,14,12, 7,19,31,26,22)( 3,15, 9, 6,17,29,28,24) ( 4,16,10, 5,18,30,27,23)$ |
$ 8, 8, 8, 8 $ | $4$ | $8$ | $( 1,15,26, 5,20,29,12,23)( 2,16,25, 6,19,30,11,24)( 3,13,27, 7,17,32,10,22) ( 4,14,28, 8,18,31, 9,21)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,20)( 2,19)( 3,17)( 4,18)( 5,23)( 6,24)( 7,22)( 8,21)( 9,28)(10,27)(11,25) (12,26)(13,32)(14,31)(15,29)(16,30)$ |
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: | $3$ | ||
Label: | 32.7 | magma: IdentifyGroup(G);
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Character table: |
2 5 4 3 3 3 3 4 4 3 3 5 1a 2a 2b 8a 8b 2c 4a 4b 8c 8d 2d 2P 1a 1a 1a 4b 4a 1a 2d 2d 4a 4b 1a 3P 1a 2a 2b 8d 8c 2c 4a 4b 8b 8a 2d 5P 1a 2a 2b 8a 8b 2c 4a 4b 8c 8d 2d 7P 1a 2a 2b 8d 8c 2c 4a 4b 8b 8a 2d 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 1 1 -1 A -A 1 -1 -1 A -A 1 X.6 1 1 -1 -A A 1 -1 -1 -A A 1 X.7 1 1 1 A A -1 -1 -1 -A -A 1 X.8 1 1 1 -A -A -1 -1 -1 A A 1 X.9 2 -2 . . . . -2 2 . . 2 X.10 2 -2 . . . . 2 -2 . . 2 X.11 4 . . . . . . . . . -4 A = -E(4) = -Sqrt(-1) = -i |
magma: CharacterTable(G);