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Magma
magma: G := TransitiveGroup(32, 38);
Group action invariants
Degree $n$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_4\times Q_8$ | ||
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: | (1,27,2,28)(3,25,4,26)(5,14,6,13)(7,16,8,15)(9,17,10,18)(11,20,12,19)(21,31,22,32)(23,29,24,30), (1,16,12,21)(2,15,11,22)(3,13,9,23)(4,14,10,24)(5,17,29,25)(6,18,30,26)(7,20,32,28)(8,19,31,27), (1,9,2,10)(3,11,4,12)(5,31,6,32)(7,29,8,30)(13,22,14,21)(15,24,16,23)(17,27,18,28)(19,26,20,25) | 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 $8$: $C_4\times C_2$ x 6, $C_2^3$, $Q_8$ x 2 $16$: $Q_8:C_2$, $C_4\times C_2^2$, $Q_8\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 7
Degree 4: $C_4$ x 4, $C_2^2$ x 7
Degree 8: $C_4\times C_2$ x 6, $C_2^3$, $Q_8$ x 2, $Q_8:C_2$ x 3
Degree 16: $C_4\times C_2^2$, $Q_8\times C_2$, $Q_8 : C_2$
Low degree siblings
There are no siblings 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 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)(17,19,18,20)(21,23,22,24) (25,27,26,28)(29,31,30,32)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 5,11,30)( 2, 6,12,29)( 3, 7,10,31)( 4, 8, 9,32)(13,20,24,27)(14,19,23,28) (15,18,21,25)(16,17,22,26)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 7,11,31)( 2, 8,12,32)( 3, 6,10,29)( 4, 5, 9,30)(13,18,24,25)(14,17,23,26) (15,19,21,28)(16,20,22,27)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 9, 2,10)( 3,11, 4,12)( 5,31, 6,32)( 7,29, 8,30)(13,22,14,21)(15,24,16,23) (17,27,18,28)(19,26,20,25)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,11)( 2,12)( 3,10)( 4, 9)( 5,30)( 6,29)( 7,31)( 8,32)(13,24)(14,23)(15,21) (16,22)(17,26)(18,25)(19,28)(20,27)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,12)( 2,11)( 3, 9)( 4,10)( 5,29)( 6,30)( 7,32)( 8,31)(13,23)(14,24)(15,22) (16,21)(17,25)(18,26)(19,27)(20,28)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,13,11,24)( 2,14,12,23)( 3,15,10,21)( 4,16, 9,22)( 5,19,30,28)( 6,20,29,27) ( 7,17,31,26)( 8,18,32,25)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,15,12,22)( 2,16,11,21)( 3,14, 9,24)( 4,13,10,23)( 5,18,29,26)( 6,17,30,25) ( 7,19,32,27)( 8,20,31,28)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,16,12,21)( 2,15,11,22)( 3,13, 9,23)( 4,14,10,24)( 5,17,29,25)( 6,18,30,26) ( 7,20,32,28)( 8,19,31,27)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,17, 2,18)( 3,20, 4,19)( 5,22, 6,21)( 7,24, 8,23)( 9,28,10,27)(11,26,12,25) (13,32,14,31)(15,30,16,29)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,19, 2,20)( 3,17, 4,18)( 5,24, 6,23)( 7,21, 8,22)( 9,25,10,26)(11,28,12,27) (13,29,14,30)(15,32,16,31)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,21,12,16)( 2,22,11,15)( 3,23, 9,13)( 4,24,10,14)( 5,25,29,17)( 6,26,30,18) ( 7,28,32,20)( 8,27,31,19)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,22,12,15)( 2,21,11,16)( 3,24, 9,14)( 4,23,10,13)( 5,26,29,18)( 6,25,30,17) ( 7,27,32,19)( 8,28,31,20)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,23,11,14)( 2,24,12,13)( 3,22,10,16)( 4,21, 9,15)( 5,27,30,20)( 6,28,29,19) ( 7,25,31,18)( 8,26,32,17)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,25, 2,26)( 3,28, 4,27)( 5,15, 6,16)( 7,14, 8,13)( 9,20,10,19)(11,18,12,17) (21,29,22,30)(23,32,24,31)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,27, 2,28)( 3,25, 4,26)( 5,14, 6,13)( 7,16, 8,15)( 9,17,10,18)(11,20,12,19) (21,31,22,32)(23,29,24,30)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,29,11, 6)( 2,30,12, 5)( 3,32,10, 8)( 4,31, 9, 7)(13,28,24,19)(14,27,23,20) (15,26,21,17)(16,25,22,18)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,31,11, 7)( 2,32,12, 8)( 3,29,10, 6)( 4,30, 9, 5)(13,25,24,18)(14,26,23,17) (15,28,21,19)(16,27,22,20)$ |
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: | 32.26 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 4A1 | 4A-1 | 4B1 | 4B-1 | 4C | 4D | 4E | 4F | 4G | 4H | 4I1 | 4I-1 | 4J1 | 4J-1 | 4K1 | 4K-1 | ||
Size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
2 P | 1A | 1A | 1A | 1A | 2B | 2B | 2B | 2B | 2C | 2A | 2A | 2A | 2C | 2C | 2A | 2C | 2A | 2A | 2C | 2C | |
Type | |||||||||||||||||||||
32.26.1a | R | ||||||||||||||||||||
32.26.1b | R | ||||||||||||||||||||
32.26.1c | R | ||||||||||||||||||||
32.26.1d | R | ||||||||||||||||||||
32.26.1e | R | ||||||||||||||||||||
32.26.1f | R | ||||||||||||||||||||
32.26.1g | R | ||||||||||||||||||||
32.26.1h | R | ||||||||||||||||||||
32.26.1i1 | C | ||||||||||||||||||||
32.26.1i2 | C | ||||||||||||||||||||
32.26.1j1 | C | ||||||||||||||||||||
32.26.1j2 | C | ||||||||||||||||||||
32.26.1k1 | C | ||||||||||||||||||||
32.26.1k2 | C | ||||||||||||||||||||
32.26.1l1 | C | ||||||||||||||||||||
32.26.1l2 | C | ||||||||||||||||||||
32.26.2a | S | ||||||||||||||||||||
32.26.2b | S | ||||||||||||||||||||
32.26.2c1 | C | ||||||||||||||||||||
32.26.2c2 | C |
magma: CharacterTable(G);