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Magma
magma: G := TransitiveGroup(32, 8);
Group action invariants
Degree $n$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $\OD_{32}$ | ||
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: | (1,18,6,23,12,28,13,30,3,19,7,22,9,26,16,32)(2,17,5,24,11,27,14,29,4,20,8,21,10,25,15,31), (1,14,9,5,3,15,12,8)(2,13,10,6,4,16,11,7)(17,32,25,22,20,30,27,23)(18,31,26,21,19,29,28,24) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $C_8$ x 2, $C_4\times C_2$ $16$: $C_8\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 8: $C_8$ x 2, $C_4\times C_2$
Degree 16: $C_8\times C_2$, $C_{16} : C_2$
Low degree siblings
16T22Siblings 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 $ | $2$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,19)(18,20)(21,23) (22,24)(25,28)(26,27)(29,32)(30,31)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,27)(26,28)(29,31)(30,32)$ |
$ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1, 5,12,14, 3, 8, 9,15)( 2, 6,11,13, 4, 7,10,16)(17,22,27,32,20,23,25,30) (18,21,28,31,19,24,26,29)$ |
$ 8, 8, 8, 8 $ | $1$ | $8$ | $( 1, 6,12,13, 3, 7, 9,16)( 2, 5,11,14, 4, 8,10,15)(17,24,27,29,20,21,25,31) (18,23,28,30,19,22,26,32)$ |
$ 8, 8, 8, 8 $ | $1$ | $8$ | $( 1, 7,12,16, 3, 6, 9,13)( 2, 8,11,15, 4, 5,10,14)(17,21,27,31,20,24,25,29) (18,22,28,32,19,23,26,30)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 9, 3,12)( 2,10, 4,11)( 5,15, 8,14)( 6,16, 7,13)(17,25,20,27)(18,26,19,28) (21,29,24,31)(22,30,23,32)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,10, 3,11)( 2, 9, 4,12)( 5,16, 8,13)( 6,15, 7,14)(17,28,20,26)(18,27,19,25) (21,32,24,30)(22,31,23,29)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,12, 3, 9)( 2,11, 4,10)( 5,14, 8,15)( 6,13, 7,16)(17,27,20,25)(18,28,19,26) (21,31,24,29)(22,32,23,30)$ |
$ 8, 8, 8, 8 $ | $1$ | $8$ | $( 1,13, 9, 6, 3,16,12, 7)( 2,14,10, 5, 4,15,11, 8)(17,29,25,24,20,31,27,21) (18,30,26,23,19,32,28,22)$ |
$ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,14, 9, 5, 3,15,12, 8)( 2,13,10, 6, 4,16,11, 7)(17,32,25,22,20,30,27,23) (18,31,26,21,19,29,28,24)$ |
$ 8, 8, 8, 8 $ | $1$ | $8$ | $( 1,16, 9, 7, 3,13,12, 6)( 2,15,10, 8, 4,14,11, 5)(17,31,25,21,20,29,27,24) (18,32,26,22,19,30,28,23)$ |
$ 16, 16 $ | $2$ | $16$ | $( 1,17, 7,21,12,27,16,31, 3,20, 6,24, 9,25,13,29)( 2,18, 8,22,11,28,15,32, 4, 19, 5,23,10,26,14,30)$ |
$ 16, 16 $ | $2$ | $16$ | $( 1,18, 6,23,12,28,13,30, 3,19, 7,22, 9,26,16,32)( 2,17, 5,24,11,27,14,29, 4, 20, 8,21,10,25,15,31)$ |
$ 16, 16 $ | $2$ | $16$ | $( 1,21,16,20, 9,29, 7,27, 3,24,13,17,12,31, 6,25)( 2,22,15,19,10,30, 8,28, 4, 23,14,18,11,32, 5,26)$ |
$ 16, 16 $ | $2$ | $16$ | $( 1,22,13,18, 9,30, 6,26, 3,23,16,19,12,32, 7,28)( 2,21,14,17,10,29, 5,25, 4, 24,15,20,11,31, 8,27)$ |
$ 16, 16 $ | $2$ | $16$ | $( 1,25, 6,31,12,17,13,24, 3,27, 7,29, 9,20,16,21)( 2,26, 5,32,11,18,14,23, 4, 28, 8,30,10,19,15,22)$ |
$ 16, 16 $ | $2$ | $16$ | $( 1,26, 7,30,12,18,16,22, 3,28, 6,32, 9,19,13,23)( 2,25, 8,29,11,17,15,21, 4, 27, 5,31,10,20,14,24)$ |
$ 16, 16 $ | $2$ | $16$ | $( 1,29,13,25, 9,24, 6,20, 3,31,16,27,12,21, 7,17)( 2,30,14,26,10,23, 5,19, 4, 32,15,28,11,22, 8,18)$ |
$ 16, 16 $ | $2$ | $16$ | $( 1,30,16,28, 9,23, 7,18, 3,32,13,26,12,22, 6,19)( 2,29,15,27,10,24, 8,17, 4, 31,14,25,11,21, 5,20)$ |
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: | 32.17 | magma: IdentifyGroup(G);
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Character table: |
2 5 4 5 4 5 5 5 4 5 5 4 5 4 4 4 4 4 4 4 1a 2a 2b 8a 8b 8c 4a 4b 4c 8d 8e 8f 16a 16b 16c 16d 16e 16f 16g 2P 1a 1a 1a 4c 4c 4c 2b 2b 2b 4a 4a 4a 8c 8b 8f 8d 8b 8c 8d 3P 1a 2a 2b 8e 8d 8f 4c 4b 4a 8b 8a 8c 16c 16d 16a 16b 16g 16h 16e 5P 1a 2a 2b 8a 8c 8b 4a 4b 4c 8f 8e 8d 16e 16f 16g 16h 16a 16b 16c 7P 1a 2a 2b 8e 8f 8d 4c 4b 4a 8c 8a 8b 16g 16h 16e 16f 16c 16d 16a 11P 1a 2a 2b 8e 8d 8f 4c 4b 4a 8b 8a 8c 16c 16d 16a 16b 16g 16h 16e 13P 1a 2a 2b 8a 8c 8b 4a 4b 4c 8f 8e 8d 16e 16f 16g 16h 16a 16b 16c X.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 X.3 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 X.5 1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 A -A -A A A -A -A X.6 1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 -A A A -A -A A A X.7 1 -1 1 A -A -A -1 1 -1 A -A A D -D -/D /D -D D /D X.8 1 -1 1 A -A -A -1 1 -1 A -A A -D D /D -/D D -D -/D X.9 1 -1 1 -A A A -1 1 -1 -A A -A -/D /D D -D /D -/D -D X.10 1 -1 1 -A A A -1 1 -1 -A A -A /D -/D -D D -/D /D D X.11 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 A A -A -A A A -A X.12 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 -A -A A A -A -A A X.13 1 1 1 A A A -1 -1 -1 -A -A -A -/D -/D D D /D /D -D X.14 1 1 1 A A A -1 -1 -1 -A -A -A /D /D -D -D -/D -/D D X.15 1 1 1 -A -A -A -1 -1 -1 A A A D D -/D -/D -D -D /D X.16 1 1 1 -A -A -A -1 -1 -1 A A A -D -D /D /D D D -/D X.17 2 . -2 . B -B C . -C -/B . /B . . . . . . . X.18 2 . -2 . -/B /B -C . C B . -B . . . . . . . X.19 2 . -2 . -B B C . -C /B . -/B . . . . . . . X.20 2 . -2 . /B -/B -C . C -B . B . . . . . . . 2 4 16h 2P 8f 3P 16f 5P 16d 7P 16b 11P 16f 13P 16d X.1 1 X.2 1 X.3 -1 X.4 -1 X.5 A X.6 -A X.7 -/D X.8 /D X.9 D X.10 -D X.11 -A X.12 A X.13 -D X.14 D X.15 /D X.16 -/D X.17 . X.18 . X.19 . X.20 . A = -E(4) = -Sqrt(-1) = -i B = -2*E(8)^3 C = 2*E(4) = 2*Sqrt(-1) = 2i D = -E(8) |
magma: CharacterTable(G);