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Magma
magma: G := TransitiveGroup(8, 36);
Group action invariants
Degree $n$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2^3:(C_7: C_3)$ | ||
CHM label: | $E(8):F_{21}$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | yes | 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: | (1,3)(2,8)(4,6)(5,7), (1,2,6,3,4,5,7), (1,8)(2,3)(4,5)(6,7), (1,2,3)(4,6,5), (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) $3$: $C_3$ $21$: $C_7:C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: None
Low degree siblings
14T11, 24T283, 28T27, 42T26Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 3, 3, 1, 1 $ | $28$ | $3$ | $(3,6,7)(4,8,5)$ |
$ 3, 3, 1, 1 $ | $28$ | $3$ | $(3,7,6)(4,5,8)$ |
$ 7, 1 $ | $24$ | $7$ | $(2,3,6,5,7,8,4)$ |
$ 7, 1 $ | $24$ | $7$ | $(2,4,8,7,5,6,3)$ |
$ 6, 2 $ | $28$ | $6$ | $(1,2)(3,4,6,8,7,5)$ |
$ 6, 2 $ | $28$ | $6$ | $(1,2)(3,5,7,8,6,4)$ |
$ 2, 2, 2, 2 $ | $7$ | $2$ | $(1,2)(3,8)(4,7)(5,6)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $168=2^{3} \cdot 3 \cdot 7$ | 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: | 168.43 | magma: IdentifyGroup(G);
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Character table: |
2 3 1 1 . . 1 1 3 3 1 1 1 . . 1 1 1 7 1 . . 1 1 . . . 1a 3a 3b 7a 7b 6a 6b 2a 2P 1a 3b 3a 7a 7b 3a 3b 1a 3P 1a 1a 1a 7b 7a 2a 2a 2a 5P 1a 3b 3a 7b 7a 6b 6a 2a 7P 1a 3a 3b 1a 1a 6a 6b 2a X.1 1 1 1 1 1 1 1 1 X.2 1 A /A 1 1 /A A 1 X.3 1 /A A 1 1 A /A 1 X.4 3 . . B /B . . 3 X.5 3 . . /B B . . 3 X.6 7 1 1 . . -1 -1 -1 X.7 7 A /A . . -/A -A -1 X.8 7 /A A . . -A -/A -1 A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7 |
magma: CharacterTable(G);