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Magma
magma: G := TransitiveGroup(16, 181);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $181$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_4\times S_4$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,11,2,12)(3,9,4,10)(5,13,6,14)(7,16,8,15), (1,10,5,12,4,8,2,9,6,11,3,7)(13,16,14,15) | 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, $C_2^2$ $6$: $S_3$ $8$: $C_4\times C_2$ $12$: $D_{6}$ $24$: $S_4$, $S_3 \times C_4$ $48$: $S_4\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 8: $S_4\times C_2$
Low degree siblings
12T53 x 2, 16T181, 24T129, 24T130, 24T167, 24T168 x 2, 24T169 x 2, 32T387Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 5,15)( 6,16)( 7,13)( 8,14)$ | |
$ 3, 3, 3, 3, 1, 1, 1, 1 $ | $8$ | $3$ | $( 3, 5,15)( 4, 6,16)( 7,13, 9)( 8,14,10)$ | |
$ 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, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 2)( 3, 4)( 5,16)( 6,15)( 7,14)( 8,13)( 9,10)(11,12)$ | |
$ 6, 6, 2, 2 $ | $8$ | $6$ | $( 1, 2)( 3, 6,15, 4, 5,16)( 7,14, 9, 8,13,10)(11,12)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 3)( 2, 4)( 5,16)( 6,15)( 7,14)( 8,13)( 9,11)(10,12)$ | |
$ 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 3, 6,15)( 2, 4, 5,16)( 7,14,12,10)( 8,13,11, 9)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 4)( 2, 3)( 5,15)( 6,16)( 7,13)( 8,14)( 9,12)(10,11)$ | |
$ 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 4, 6,16)( 2, 3, 5,15)( 7,13,12, 9)( 8,14,11,10)$ | |
$ 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 7, 2, 8)( 3,10, 4, 9)( 5,11, 6,12)(13,15,14,16)$ | |
$ 12, 4 $ | $8$ | $12$ | $( 1, 7,15,11, 6,13, 2, 8,16,12, 5,14)( 3,10, 4, 9)$ | |
$ 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 7,15,10)( 2, 8,16, 9)( 3,11, 6,13)( 4,12, 5,14)$ | |
$ 4, 4, 4, 4 $ | $3$ | $4$ | $( 1, 7, 2, 8)( 3,14, 4,13)( 5,11, 6,12)( 9,15,10,16)$ | |
$ 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 8, 2, 7)( 3, 9, 4,10)( 5,12, 6,11)(13,16,14,15)$ | |
$ 12, 4 $ | $8$ | $12$ | $( 1, 8,15,12, 6,14, 2, 7,16,11, 5,13)( 3, 9, 4,10)$ | |
$ 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 8,15, 9)( 2, 7,16,10)( 3,12, 6,14)( 4,11, 5,13)$ | |
$ 4, 4, 4, 4 $ | $3$ | $4$ | $( 1, 8, 2, 7)( 3,13, 4,14)( 5,12, 6,11)( 9,16,10,15)$ | |
$ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,11, 2,12)( 3, 9, 4,10)( 5, 7, 6, 8)(13,16,14,15)$ | |
$ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,12, 2,11)( 3,10, 4, 9)( 5, 8, 6, 7)(13,15,14,16)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $96=2^{5} \cdot 3$ | 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: | not nilpotent | ||
Label: | 96.186 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 2D | 2E | 3A | 4A1 | 4A-1 | 4B1 | 4B-1 | 4C | 4D | 4E1 | 4E-1 | 4F1 | 4F-1 | 6A | 12A1 | 12A-1 | ||
Size | 1 | 1 | 3 | 3 | 6 | 6 | 8 | 1 | 1 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | |
2 P | 1A | 1A | 1A | 1A | 1A | 1A | 3A | 2A | 2A | 2A | 2A | 2B | 2C | 2B | 2A | 2A | 2C | 3A | 6A | 6A | |
3 P | 1A | 2A | 2B | 2C | 2D | 2E | 1A | 4A-1 | 4A1 | 4B-1 | 4B1 | 4D | 4E-1 | 4C | 4F-1 | 4F1 | 4E1 | 2A | 4A1 | 4A-1 | |
Type | |||||||||||||||||||||
96.186.1a | R | ||||||||||||||||||||
96.186.1b | R | ||||||||||||||||||||
96.186.1c | R | ||||||||||||||||||||
96.186.1d | R | ||||||||||||||||||||
96.186.1e1 | C | ||||||||||||||||||||
96.186.1e2 | C | ||||||||||||||||||||
96.186.1f1 | C | ||||||||||||||||||||
96.186.1f2 | C | ||||||||||||||||||||
96.186.2a | R | ||||||||||||||||||||
96.186.2b | R | ||||||||||||||||||||
96.186.2c1 | C | ||||||||||||||||||||
96.186.2c2 | C | ||||||||||||||||||||
96.186.3a | R | ||||||||||||||||||||
96.186.3b | R | ||||||||||||||||||||
96.186.3c | R | ||||||||||||||||||||
96.186.3d | R | ||||||||||||||||||||
96.186.3e1 | C | ||||||||||||||||||||
96.186.3e2 | C | ||||||||||||||||||||
96.186.3f1 | C | ||||||||||||||||||||
96.186.3f2 | C |
magma: CharacterTable(G);