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Magma
magma: G := TransitiveGroup(12, 205);
Group action invariants
| Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $205$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2^4:(C_3\times S_4)$ | ||
| CHM label: | $[E(4)^{3}:3:2]3$ | ||
| 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: | (3,12)(6,9), (3,9)(6,12), (2,8,11)(3,6,12)(4,7,10), (1,10)(2,5)(6,9), (1,5,9)(2,6,10)(3,7,11)(4,8,12) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $18$: $S_3\times C_3$ $24$: $S_4$ $72$: 12T45 $288$: $A_4\wr C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 4: None
Degree 6: None
Low degree siblings
18T269, 18T270, 24T2765, 24T2767, 24T2812, 24T2823, 24T2829, 24T2831, 32T96681, 36T1610, 36T1615, 36T1617, 36T1724, 36T1725, 36T1755, 36T1756, 36T1757, 36T1896 x 2, 36T1902, 36T1903, 36T1904, 36T1905, 36T1940, 36T1941, 36T1942 x 2Siblings 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 $ | $1$ | $1$ | $()$ | |
| $ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 3,12)( 6, 9)$ | |
| $ 2, 2, 2, 2, 1, 1, 1, 1 $ | $18$ | $2$ | $( 1, 4)( 3,12)( 6, 9)( 7,10)$ | |
| $ 2, 2, 2, 2, 1, 1, 1, 1 $ | $9$ | $2$ | $( 1, 4)( 3, 6)( 7,10)( 9,12)$ | |
| $ 2, 2, 2, 2, 2, 2 $ | $18$ | $2$ | $( 1, 4)( 2,11)( 3,12)( 5, 8)( 6, 9)( 7,10)$ | |
| $ 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$ | |
| $ 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 4)( 2, 8)( 3,12)( 5,11)( 6, 9)( 7,10)$ | |
| $ 3, 3, 3, 1, 1, 1 $ | $128$ | $3$ | $( 4, 7,10)( 5,11, 8)( 6, 9,12)$ | |
| $ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $24$ | $2$ | $( 5, 8)( 7,10)( 9,12)$ | |
| $ 4, 2, 2, 1, 1, 1, 1 $ | $72$ | $4$ | $( 3,12, 6, 9)( 5, 8)( 7,10)$ | |
| $ 4, 4, 2, 1, 1 $ | $72$ | $4$ | $( 2, 8,11, 5)( 3,12, 6, 9)( 7,10)$ | |
| $ 4, 4, 4 $ | $24$ | $4$ | $( 1, 7, 4,10)( 2, 8,11, 5)( 3,12, 6, 9)$ | |
| $ 3, 3, 3, 3 $ | $16$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ | |
| $ 6, 6 $ | $48$ | $6$ | $( 1, 5, 6,10, 2, 9)( 3, 7,11,12, 4, 8)$ | |
| $ 6, 6 $ | $96$ | $6$ | $( 1,11, 3,10, 2, 9)( 4, 5,12, 7, 8, 6)$ | |
| $ 3, 3, 3, 3 $ | $32$ | $3$ | $( 1,11, 9)( 2, 3,10)( 4, 5, 6)( 7, 8,12)$ | |
| $ 6, 3, 3 $ | $96$ | $6$ | $( 1, 8, 9)( 2, 6, 7,11, 3,10)( 4, 5,12)$ | |
| $ 12 $ | $96$ | $12$ | $( 1, 8, 6, 7,11,12, 4, 5, 3,10, 2, 9)$ | |
| $ 3, 3, 3, 3 $ | $16$ | $3$ | $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$ | |
| $ 6, 6 $ | $48$ | $6$ | $( 1, 6, 2,10, 9, 5)( 3,11, 7,12, 8, 4)$ | |
| $ 3, 3, 3, 3 $ | $32$ | $3$ | $( 1,12, 5)( 2, 4, 6)( 3, 8, 7)( 9,11,10)$ | |
| $ 6, 6 $ | $96$ | $6$ | $( 1, 3, 8, 7,12, 5)( 2, 4, 9,11,10, 6)$ | |
| $ 6, 3, 3 $ | $96$ | $6$ | $( 1,12, 5)( 2, 7, 3,11,10, 6)( 4, 9, 8)$ | |
| $ 12 $ | $96$ | $12$ | $( 1, 3,11,10, 9, 8, 4, 6, 2, 7,12, 5)$ |
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: | 1152.157858 | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| Type |
magma: CharacterTable(G);