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Magma
magma: G := TransitiveGroup(32, 6);
Group action invariants
Degree $n$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $6$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_4.C_2^3$ | ||
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: | (1,10,4,12)(2,9,3,11)(5,14,8,15)(6,13,7,16)(17,27,19,26)(18,28,20,25)(21,32,24,29)(22,31,23,30), (1,17,4,19)(2,18,3,20)(5,30,8,31)(6,29,7,32)(9,28,11,25)(10,27,12,26)(13,21,16,24)(14,22,15,23), (1,20)(2,19)(3,17)(4,18)(5,32)(6,31)(7,30)(8,29)(9,26)(10,25)(11,27)(12,28)(13,23)(14,24)(15,21)(16,22), (1,6,4,7)(2,5,3,8)(9,15,11,14)(10,16,12,13)(17,32,19,29)(18,31,20,30)(21,26,24,27)(22,25,23,28) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 15 $4$: $C_2^2$ x 35 $8$: $C_2^3$ x 15 $16$: $C_2^4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 15
Degree 4: $C_2^2$ x 35
Degree 8: $C_2^3$ x 15
Degree 16: $C_2^4$, $(C_2 \times Q_8):C_2$ x 5
Low degree siblings
16T20 x 5Siblings 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 $ | $2$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(21,23) (22,24)(25,27)(26,28)(29,31)(30,32)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 5, 4, 8)( 2, 6, 3, 7)( 9,16,11,13)(10,15,12,14)(17,30,19,31)(18,29,20,32) (21,28,24,25)(22,27,23,26)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 6, 4, 7)( 2, 5, 3, 8)( 9,15,11,14)(10,16,12,13)(17,32,19,29)(18,31,20,30) (21,26,24,27)(22,25,23,28)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 9, 4,11)( 2,10, 3,12)( 5,13, 8,16)( 6,14, 7,15)(17,25,19,28)(18,26,20,27) (21,30,24,31)(22,29,23,32)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,10, 4,12)( 2, 9, 3,11)( 5,14, 8,15)( 6,13, 7,16)(17,27,19,26)(18,28,20,25) (21,32,24,29)(22,31,23,30)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,13, 4,16)( 2,14, 3,15)( 5,11, 8, 9)( 6,12, 7,10)(17,24,19,21)(18,23,20,22) (25,30,28,31)(26,29,27,32)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,14, 4,15)( 2,13, 3,16)( 5,12, 8,10)( 6,11, 7, 9)(17,22,19,23)(18,21,20,24) (25,32,28,29)(26,31,27,30)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,17, 4,19)( 2,18, 3,20)( 5,30, 8,31)( 6,29, 7,32)( 9,28,11,25)(10,27,12,26) (13,21,16,24)(14,22,15,23)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1,18)( 2,17)( 3,19)( 4,20)( 5,29)( 6,30)( 7,31)( 8,32)( 9,27)(10,28)(11,26) (12,25)(13,22)(14,21)(15,24)(16,23)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,21, 4,24)( 2,22, 3,23)( 5,25, 8,28)( 6,26, 7,27)( 9,30,11,31)(10,29,12,32) (13,19,16,17)(14,20,15,18)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1,22)( 2,21)( 3,24)( 4,23)( 5,26)( 6,25)( 7,28)( 8,27)( 9,29)(10,30)(11,32) (12,31)(13,20)(14,19)(15,17)(16,18)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,25, 4,28)( 2,26, 3,27)( 5,24, 8,21)( 6,23, 7,22)( 9,17,11,19)(10,18,12,20) (13,30,16,31)(14,29,15,32)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1,26)( 2,25)( 3,28)( 4,27)( 5,23)( 6,24)( 7,21)( 8,22)( 9,18)(10,17)(11,20) (12,19)(13,29)(14,30)(15,31)(16,32)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,29, 4,32)( 2,30, 3,31)( 5,20, 8,18)( 6,19, 7,17)( 9,23,11,22)(10,24,12,21) (13,27,16,26)(14,28,15,25)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1,30)( 2,29)( 3,32)( 4,31)( 5,19)( 6,20)( 7,18)( 8,17)( 9,24)(10,23)(11,21) (12,22)(13,28)(14,27)(15,26)(16,25)$ |
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: | 32.50 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | ||
Size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
2 P | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | 2A | 2A | 2A | 2A | 2A | 2A | |
Type | ||||||||||||||||||
32.50.1a | R | |||||||||||||||||
32.50.1b | R | |||||||||||||||||
32.50.1c | R | |||||||||||||||||
32.50.1d | R | |||||||||||||||||
32.50.1e | R | |||||||||||||||||
32.50.1f | R | |||||||||||||||||
32.50.1g | R | |||||||||||||||||
32.50.1h | R | |||||||||||||||||
32.50.1i | R | |||||||||||||||||
32.50.1j | R | |||||||||||||||||
32.50.1k | R | |||||||||||||||||
32.50.1l | R | |||||||||||||||||
32.50.1m | R | |||||||||||||||||
32.50.1n | R | |||||||||||||||||
32.50.1o | R | |||||||||||||||||
32.50.1p | R | |||||||||||||||||
32.50.4a | S |
magma: CharacterTable(G);