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Magma
magma: G := TransitiveGroup(32, 19);
Group action invariants
Degree $n$: | $32$ | 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: | $C_2^3:C_4$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $32$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,29,11,7)(2,30,12,8)(3,31,10,6)(4,32,9,5)(13,15,14,16)(17,25,18,26)(19,27,20,28)(21,23,22,24), (1,23,5,27)(2,24,6,28)(3,22,7,25)(4,21,8,26)(9,16,30,20)(10,15,29,19)(11,13,32,18)(12,14,31,17) | 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_2^3 : C_4 $ x 2, $C_2^3: C_4$, $C_2^3: C_4$
Degree 16: $C_2^2 : C_4$, 16T33 x 2, 16T52, 16T53
Low degree siblings
8T19 x 2, 8T20, 8T21, 16T33 x 2, 16T52, 16T53Siblings 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, 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 $ | $2$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,23)(14,24)(15,22)(16,21)(17,28) (18,27)(19,25)(20,26)(29,30)(31,32)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 3, 2, 4)( 5,30, 6,29)( 7,32, 8,31)( 9,11,10,12)(13,20,23,26)(14,19,24,25) (15,18,22,27)(16,17,21,28)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 4, 2, 3)( 5,29, 6,30)( 7,31, 8,32)( 9,12,10,11)(13,26,23,20)(14,25,24,19) (15,27,22,18)(16,28,21,17)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,30)(10,29)(11,32)(12,31)(13,18)(14,17)(15,19) (16,20)(21,26)(22,25)(23,27)(24,28)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,29)(10,30)(11,31)(12,32)(13,27)(14,28)(15,25) (16,26)(17,24)(18,23)(19,22)(20,21)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,12)( 2,11)( 3, 9)( 4,10)( 5,31)( 6,32)( 7,30)( 8,29)(13,24)(14,23)(15,21) (16,22)(17,27)(18,28)(19,26)(20,25)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,13,31,28)( 2,14,32,27)( 3,15,30,26)( 4,16,29,25)( 5,18,12,24)( 6,17,11,23) ( 7,19, 9,21)( 8,20,10,22)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,14, 5,17)( 2,13, 6,18)( 3,16, 7,20)( 4,15, 8,19)( 9,22,30,25)(10,21,29,26) (11,24,32,28)(12,23,31,27)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,15,12,21)( 2,16,11,22)( 3,14, 9,23)( 4,13,10,24)( 5,26,31,19)( 6,25,32,20) ( 7,27,30,17)( 8,28,29,18)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1,16)( 2,15)( 3,13)( 4,14)( 5,25)( 6,26)( 7,28)( 8,27)( 9,24)(10,23)(11,21) (12,22)(17,29)(18,30)(19,32)(20,31)$ |
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: | $3$ | ||
Label: | 32.6 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 2D | 2E | 4A | 4B1 | 4B-1 | 4C1 | 4C-1 | ||
Size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | |
2 P | 1A | 1A | 1A | 1A | 1A | 1A | 2C | 2D | 2A | 2C | 2D | |
Type | ||||||||||||
32.6.1a | R | |||||||||||
32.6.1b | R | |||||||||||
32.6.1c | R | |||||||||||
32.6.1d | R | |||||||||||
32.6.1e1 | C | |||||||||||
32.6.1e2 | C | |||||||||||
32.6.1f1 | C | |||||||||||
32.6.1f2 | C | |||||||||||
32.6.2a | R | |||||||||||
32.6.2b | R | |||||||||||
32.6.4a | R |
magma: CharacterTable(G);