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Magma
magma: G := TransitiveGroup(16, 19);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $19$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_4 \times D_4$ | ||
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: | (1,11,2,12)(3,9,4,10)(5,7,6,8)(13,15,14,16), (1,6,2,5)(3,16,4,15)(7,11,8,12)(9,13,10,14), (1,13,16,11)(2,14,15,12)(3,7,5,9)(4,8,6,10) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_4$ x 4, $C_2^2$ x 7 $8$: $D_{4}$ x 2, $C_4\times C_2$ x 6, $C_2^3$ $16$: $D_4\times C_2$, $Q_8:C_2$, $C_4\times C_2^2$ 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 2
Degree 8: $C_4\times C_2$, $D_4\times C_2$, $Q_8:C_2$
Low degree siblings
16T19 x 3, 32T5Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | 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 $ | $2$ | $2$ | $( 7,10)( 8, 9)(11,14)(12,13)$ | |
$ 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 $ | $2$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 9)( 8,10)(11,13)(12,14)(15,16)$ | |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5,15, 6,16)( 7,11, 8,12)( 9,13,10,14)$ | |
$ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 3, 2, 4)( 5,15, 6,16)( 7,14, 8,13)( 9,12,10,11)$ | |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 4, 2, 3)( 5,16, 6,15)( 7,12, 8,11)( 9,14,10,13)$ | |
$ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 4, 2, 3)( 5,16, 6,15)( 7,13, 8,14)( 9,11,10,12)$ | |
$ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 5, 2, 6)( 3,15, 4,16)( 7,12, 8,11)( 9,14,10,13)$ | |
$ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 6, 2, 5)( 3,16, 4,15)( 7,11, 8,12)( 9,13,10,14)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 7)( 2, 8)( 3,14)( 4,13)( 5,12)( 6,11)( 9,16)(10,15)$ | |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 7,15,10)( 2, 8,16, 9)( 3,14, 6,11)( 4,13, 5,12)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 8)( 2, 7)( 3,13)( 4,14)( 5,11)( 6,12)( 9,15)(10,16)$ | |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 8,15, 9)( 2, 7,16,10)( 3,13, 6,12)( 4,14, 5,11)$ | |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,11,16,13)( 2,12,15,14)( 3, 9, 5, 7)( 4,10, 6, 8)$ | |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,11, 2,12)( 3, 9, 4,10)( 5, 7, 6, 8)(13,15,14,16)$ | |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,12,16,14)( 2,11,15,13)( 3,10, 5, 8)( 4, 9, 6, 7)$ | |
$ 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,12, 2,11)( 3,10, 4, 9)( 5, 8, 6, 7)(13,16,14,15)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,15)( 2,16)( 3, 6)( 4, 5)( 7,10)( 8, 9)(11,14)(12,13)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,16)( 2,15)( 3, 5)( 4, 6)( 7, 9)( 8,10)(11,13)(12,14)$ |
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: | 32.25 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A1 | 4A-1 | 4B1 | 4B-1 | 4C | 4D | 4E1 | 4E-1 | 4F1 | 4F-1 | 4G1 | 4G-1 | ||
Size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
2 P | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2B | 2B | 2B | 2B | 2B | 2B | 2C | 2B | 2B | 2A | 2A | 2C | |
Type | |||||||||||||||||||||
32.25.1a | R | ||||||||||||||||||||
32.25.1b | R | ||||||||||||||||||||
32.25.1c | R | ||||||||||||||||||||
32.25.1d | R | ||||||||||||||||||||
32.25.1e | R | ||||||||||||||||||||
32.25.1f | R | ||||||||||||||||||||
32.25.1g | R | ||||||||||||||||||||
32.25.1h | R | ||||||||||||||||||||
32.25.1i1 | C | ||||||||||||||||||||
32.25.1i2 | C | ||||||||||||||||||||
32.25.1j1 | C | ||||||||||||||||||||
32.25.1j2 | C | ||||||||||||||||||||
32.25.1k1 | C | ||||||||||||||||||||
32.25.1k2 | C | ||||||||||||||||||||
32.25.1l1 | C | ||||||||||||||||||||
32.25.1l2 | C | ||||||||||||||||||||
32.25.2a | R | ||||||||||||||||||||
32.25.2b | R | ||||||||||||||||||||
32.25.2c1 | C | ||||||||||||||||||||
32.25.2c2 | C |
magma: CharacterTable(G);