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Magma
magma: G := TransitiveGroup(16, 1194);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $1194$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2^7:C_8$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,12,5,16,9,3,14,7)(2,11,6,15,10,4,13,8), (1,14,10,5,2,13,9,6)(3,15,11,7,4,16,12,8) | 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, $C_8$ x 2, $C_4\times C_2$ $16$: $C_8:C_2$, $C_2^2:C_4$, $C_8\times C_2$ $32$: $(C_8:C_2):C_2$, $C_2^3 : C_4 $, $C_2^2 : C_8$ $64$: $((C_8 : C_2):C_2):C_2$ x 2, 16T84 $128$: 16T228 $256$: 16T565 $512$: 16T817 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 8: $C_8$
Low degree siblings
16T1194 x 7, 16T1216 x 8, 32T36186 x 16, 32T36187 x 16, 32T36188 x 16, 32T36189 x 16, 32T36190 x 16, 32T36191 x 8, 32T36192 x 32, 32T36193 x 16, 32T36194 x 16, 32T36195 x 32, 32T36196 x 8, 32T36197 x 16, 32T36198 x 16, 32T36199 x 4, 32T36200 x 8, 32T36201 x 8, 32T36202 x 4, 32T36203 x 8, 32T36350 x 16, 32T36351 x 8, 32T36352 x 8, 32T36353 x 16, 32T36354 x 4, 32T36355 x 4, 32T50842 x 4Siblings 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, 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 3, 4)( 7, 8)(11,12)(15,16)$ | |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 5, 6)( 7, 8)(13,14)(15,16)$ | |
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 7, 8)(15,16)$ | |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $4$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 9,10)(11,12)(13,14)$ | |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 9,10)(11,12)(13,14)(15,16)$ | |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 4)( 7, 8)( 9,10)(13,14)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $8$ | $2$ | $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14)( 6,13)( 7,15)( 8,16)$ | |
$ 4, 4, 4, 4 $ | $16$ | $4$ | $( 1,10, 2, 9)( 3,11, 4,12)( 5,13, 6,14)( 7,15, 8,16)$ | |
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $(11,12)(15,16)$ | |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(13,14)$ | |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 5, 6)( 7, 8)(11,12)(13,14)$ | |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 1, 2)( 3, 4)( 9,10)(15,16)$ | |
$ 4, 4, 2, 2, 2, 2 $ | $32$ | $4$ | $( 1, 9)( 2,10)( 3,12, 4,11)( 5,14)( 6,13)( 7,16, 8,15)$ | |
$ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1, 5, 9,14)( 2, 6,10,13)( 3, 7,12,16)( 4, 8,11,15)$ | |
$ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1,14, 9, 5)( 2,13,10, 6)( 3,16,12, 7)( 4,15,11, 8)$ | |
$ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1, 5, 9,14)( 2, 6,10,13)( 3, 7,11,16)( 4, 8,12,15)$ | |
$ 4, 4, 4, 4 $ | $32$ | $4$ | $( 1,14, 9, 5)( 2,13,10, 6)( 3,16,12, 8)( 4,15,11, 7)$ | |
$ 8, 8 $ | $64$ | $8$ | $( 1,12, 5,16, 9, 3,14, 7)( 2,11, 6,15,10, 4,13, 8)$ | |
$ 8, 8 $ | $64$ | $8$ | $( 1, 3, 5, 7, 9,12,14,16)( 2, 4, 6, 8,10,11,13,15)$ | |
$ 8, 8 $ | $64$ | $8$ | $( 1,16,14,12, 9, 7, 5, 3)( 2,15,13,11,10, 8, 6, 4)$ | |
$ 8, 8 $ | $64$ | $8$ | $( 1, 7,14, 3, 9,16, 5,12)( 2, 8,13, 4,10,15, 6,11)$ | |
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $(13,14)(15,16)$ | |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$ | |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 4)( 7, 8)(11,12)(13,14)$ | |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 1, 2)( 5, 6)( 9,10)(15,16)$ | |
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 7, 8)(13,14)$ | |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $8$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 9,10)(11,12)(15,16)$ | |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 3, 4)(11,12)(13,14)(15,16)$ | |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $2$ | $( 1, 2)( 5, 6)( 7, 8)( 9,10)$ | |
$ 4, 4, 2, 2, 2, 2 $ | $32$ | $4$ | $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14, 6,13)( 7,16, 8,15)$ | |
$ 4, 4, 2, 2, 2, 2 $ | $32$ | $4$ | $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14, 6,13)( 7,15, 8,16)$ | |
$ 8, 8 $ | $64$ | $8$ | $( 1, 5, 9,14, 2, 6,10,13)( 3, 7,12,16, 4, 8,11,15)$ | |
$ 8, 8 $ | $64$ | $8$ | $( 1,14,10, 6, 2,13, 9, 5)( 3,16,11, 8, 4,15,12, 7)$ | |
$ 8, 8 $ | $64$ | $8$ | $( 1,12, 5,16,10, 4,13, 7)( 2,11, 6,15, 9, 3,14, 8)$ | |
$ 8, 8 $ | $64$ | $8$ | $( 1, 3, 5, 7, 9,12,14,15)( 2, 4, 6, 8,10,11,13,16)$ | |
$ 8, 8 $ | $64$ | $8$ | $( 1,16,13,12, 9, 7, 5, 3)( 2,15,14,11,10, 8, 6, 4)$ | |
$ 8, 8 $ | $64$ | $8$ | $( 1, 7,14, 4,10,15, 5,12)( 2, 8,13, 3, 9,16, 6,11)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $1024=2^{10}$ | 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: | $7$ | ||
Label: | 1024.dgi | magma: IdentifyGroup(G);
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Character table: | 40 x 40 character table |
magma: CharacterTable(G);