Show commands:
Magma
magma: G := TransitiveGroup(8, 47);
Group action invariants
| Degree $n$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $47$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_4\wr C_2$ | ||
| CHM label: | $[S(4)^{2}]2$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,2,3,8), (2,3), (1,5)(2,6)(3,7)(4,8) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $D_{4}$ $72$: $C_3^2:D_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: None
Low degree siblings
12T200, 12T201, 12T202, 12T203, 16T1292, 16T1294, 16T1295, 16T1296, 18T272, 18T273, 18T274, 18T275, 24T2803, 24T2804, 24T2805, 24T2806, 24T2807, 24T2808, 24T2809, 24T2810, 24T2821, 24T2826, 32T96692, 32T96694, 32T96695, 32T96696, 36T1758, 36T1759, 36T1760, 36T1761, 36T1762, 36T1763, 36T1764, 36T1765, 36T1766, 36T1767, 36T1768, 36T1769, 36T1943, 36T1944, 36T1945, 36T1946Siblings 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 |
| $1^{8}$ | $1$ | $1$ | $()$ | |
| $2^{2},1^{4}$ | $6$ | $2$ | $(1,3)(2,8)$ | |
| $2^{4}$ | $9$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | |
| $3,1^{5}$ | $16$ | $3$ | $(2,3,8)$ | |
| $3,2^{2},1$ | $48$ | $6$ | $(2,3,8)(4,6)(5,7)$ | |
| $3^{2},1^{2}$ | $64$ | $3$ | $(2,3,8)(5,7,6)$ | |
| $2,1^{6}$ | $12$ | $2$ | $(3,8)$ | |
| $4,1^{4}$ | $12$ | $4$ | $(1,3,2,8)$ | |
| $2^{3},1^{2}$ | $36$ | $2$ | $(3,8)(4,6)(5,7)$ | |
| $4,2^{2}$ | $36$ | $4$ | $(1,3,2,8)(4,6)(5,7)$ | |
| $3,2,1^{3}$ | $96$ | $6$ | $(3,8)(5,7,6)$ | |
| $4,3,1$ | $96$ | $12$ | $(1,3,2,8)(5,7,6)$ | |
| $2^{2},1^{4}$ | $36$ | $2$ | $(3,8)(6,7)$ | |
| $4,2,1^{2}$ | $72$ | $4$ | $(1,3,2,8)(6,7)$ | |
| $4^{2}$ | $36$ | $4$ | $(1,3,2,8)(4,6,5,7)$ | |
| $2^{4}$ | $24$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | |
| $4^{2}$ | $72$ | $4$ | $(1,5,3,7)(2,6,8,4)$ | |
| $6,2$ | $192$ | $6$ | $(1,5)(2,6,3,7,8,4)$ | |
| $4,2^{2}$ | $144$ | $4$ | $(1,5)(2,6)(3,7,8,4)$ | |
| $8$ | $144$ | $8$ | $(1,5,3,7,2,6,8,4)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $1152=2^{7} \cdot 3^{2}$ | 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: | 1152.157849 | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| Type |
magma: CharacterTable(G);