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Magma
magma: G := TransitiveGroup(32, 26);
Group action invariants
Degree $n$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $D_8:C_2$ | ||
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: | (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,19)(14,20)(15,18)(16,17)(21,26)(22,25)(23,27)(24,28)(29,32)(30,31), (1,27,29,24,4,26,31,22)(2,28,30,23,3,25,32,21)(5,14,10,19,7,16,11,18)(6,13,9,20,8,15,12,17), (1,16)(2,15)(3,13)(4,14)(5,27)(6,28)(7,26)(8,25)(9,21)(10,22)(11,24)(12,23)(17,32)(18,31)(19,29)(20,30) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $8$: $D_{4}$ x 2, $C_2^3$ $16$: $D_4\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 7
Degree 4: $C_2^2$ x 7, $D_{4}$ x 4
Degree 8: $C_2^3$, $D_4$ x 2, $D_4\times C_2$ x 4
Degree 16: $D_4\times C_2$, 16T44 x 2, 16T47
Low degree siblings
16T44 x 2, 16T47Siblings 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 $ | $4$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,11)(10,12)(13,19)(14,20)(15,18)(16,17)(21,26) (22,25)(23,27)(24,28)(29,32)(30,31)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,23) (22,24)(25,28)(26,27)(29,31)(30,32)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1, 5, 4, 7)( 2, 6, 3, 8)( 9,30,12,32)(10,29,11,31)(13,21,15,23)(14,22,16,24) (17,28,20,25)(18,27,19,26)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 6, 4, 8)( 2, 5, 3, 7)( 9,31,12,29)(10,32,11,30)(13,26,15,27)(14,25,16,28) (17,24,20,22)(18,23,19,21)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 8, 4, 6)( 2, 7, 3, 5)( 9,29,12,31)(10,30,11,32)(13,27,15,26)(14,28,16,25) (17,22,20,24)(18,21,19,23)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,32)( 6,31)( 7,30)( 8,29)(13,22)(14,21)(15,24) (16,23)(17,26)(18,25)(19,28)(20,27)$ | |
$ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,13,31,17, 4,15,29,20)( 2,14,32,18, 3,16,30,19)( 5,25,11,23, 7,28,10,21) ( 6,26,12,24, 8,27, 9,22)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $4$ | $2$ | $( 1,14)( 2,13)( 3,15)( 4,16)( 5,26)( 6,25)( 7,27)( 8,28)( 9,23)(10,24)(11,22) (12,21)(17,30)(18,29)(19,31)(20,32)$ | |
$ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,15,31,20, 4,13,29,17)( 2,16,32,19, 3,14,30,18)( 5,28,11,21, 7,25,10,23) ( 6,27,12,22, 8,26, 9,24)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $4$ | $4$ | $( 1,21, 4,23)( 2,22, 3,24)( 5,17, 7,20)( 6,18, 8,19)( 9,16,12,14)(10,15,11,13) (25,31,28,29)(26,32,27,30)$ | |
$ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,22,31,26, 4,24,29,27)( 2,21,32,25, 3,23,30,28)( 5,18,11,16, 7,19,10,14) ( 6,17,12,15, 8,20, 9,13)$ | |
$ 8, 8, 8, 8 $ | $2$ | $8$ | $( 1,24,31,27, 4,22,29,26)( 2,23,32,28, 3,21,30,25)( 5,19,11,14, 7,18,10,16) ( 6,20,12,13, 8,17, 9,15)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,29, 4,31)( 2,30, 3,32)( 5,10, 7,11)( 6, 9, 8,12)(13,20,15,17)(14,19,16,18) (21,28,23,25)(22,27,24,26)$ |
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: | 32.42 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 2D | 4A1 | 4A-1 | 4B | 4C | 4D | 8A1 | 8A3 | 8B1 | 8B-1 | ||
Size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | |
2 P | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | 2A | 4B | 4B | 4B | 4B | |
Type | |||||||||||||||
32.42.1a | R | ||||||||||||||
32.42.1b | R | ||||||||||||||
32.42.1c | R | ||||||||||||||
32.42.1d | R | ||||||||||||||
32.42.1e | R | ||||||||||||||
32.42.1f | R | ||||||||||||||
32.42.1g | R | ||||||||||||||
32.42.1h | R | ||||||||||||||
32.42.2a | R | ||||||||||||||
32.42.2b | R | ||||||||||||||
32.42.2c1 | C | ||||||||||||||
32.42.2c2 | C | ||||||||||||||
32.42.2c3 | C | ||||||||||||||
32.42.2c4 | C |
magma: CharacterTable(G);