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Magma
magma: G := TransitiveGroup(36, 21);
Group action invariants
Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $S_3\times A_4$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $12$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,13,27,3,16,25)(2,14,28,4,15,26)(5,9,31,33,19,21)(6,10,32,34,20,22)(7,12,29,36,17,23)(8,11,30,35,18,24), (1,34,7)(2,33,8)(3,36,6)(4,35,5)(9,17,15,10,18,16)(11,20,14,12,19,13)(21,29,28,22,30,27)(23,31,25,24,32,26) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $S_3$, $C_6$ $12$: $A_4$ $18$: $S_3\times C_3$ $24$: $A_4\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: None
Degree 6: $C_6$, $S_3$, $A_4$, $S_3\times C_3$, $A_4\times C_2$
Degree 9: $S_3\times C_3$
Degree 12: $A_4 \times C_2$
Degree 18: $S_3 \times C_3$, 18T31, 18T32
Low degree siblings
12T43, 18T31, 18T32, 24T78, 24T83, 36T50, 36T51Siblings 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 |
$1^{36}$ | $1$ | $1$ | $()$ | |
$2^{12},1^{12}$ | $3$ | $2$ | $( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30) (31,32)$ | |
$2^{18}$ | $3$ | $2$ | $( 1, 3)( 2, 4)( 5,33)( 6,34)( 7,36)( 8,35)( 9,19)(10,20)(11,18)(12,17)(13,16) (14,15)(21,31)(22,32)(23,29)(24,30)(25,27)(26,28)$ | |
$2^{18}$ | $9$ | $2$ | $( 1, 3)( 2, 4)( 5,33)( 6,34)( 7,36)( 8,35)( 9,20)(10,19)(11,17)(12,18)(13,15) (14,16)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)$ | |
$3^{12}$ | $2$ | $3$ | $( 1, 7,34)( 2, 8,33)( 3, 6,36)( 4, 5,35)( 9,15,18)(10,16,17)(11,14,19) (12,13,20)(21,28,30)(22,27,29)(23,25,32)(24,26,31)$ | |
$6^{4},3^{4}$ | $6$ | $6$ | $( 1, 7,34)( 2, 8,33)( 3, 6,36)( 4, 5,35)( 9,16,18,10,15,17)(11,13,19,12,14,20) (21,27,30,22,28,29)(23,26,32,24,25,31)$ | |
$3^{12}$ | $8$ | $3$ | $( 1, 9,29)( 2,10,30)( 3,11,32)( 4,12,31)( 5,13,24)( 6,14,23)( 7,15,22) ( 8,16,21)(17,28,33)(18,27,34)(19,25,36)(20,26,35)$ | |
$6^{6}$ | $12$ | $6$ | $( 1,11,27,36,15,23)( 2,12,28,35,16,24)( 3, 9,25,34,14,22)( 4,10,26,33,13,21) ( 5,17,31, 8,20,30)( 6,18,32, 7,19,29)$ | |
$3^{12}$ | $4$ | $3$ | $( 1,15,28)( 2,16,27)( 3,14,26)( 4,13,25)( 5,20,32)( 6,19,31)( 7,18,30) ( 8,17,29)( 9,21,34)(10,22,33)(11,24,36)(12,23,35)$ | |
$3^{12}$ | $8$ | $3$ | $( 1,21,18)( 2,22,17)( 3,24,19)( 4,23,20)( 5,25,12)( 6,26,11)( 7,28, 9) ( 8,27,10)(13,35,32)(14,36,31)(15,34,30)(16,33,29)$ | |
$6^{6}$ | $12$ | $6$ | $( 1,23,15,36,27,11)( 2,24,16,35,28,12)( 3,22,14,34,25, 9)( 4,21,13,33,26,10) ( 5,30,20, 8,31,17)( 6,29,19, 7,32,18)$ | |
$3^{12}$ | $4$ | $3$ | $( 1,27,16)( 2,28,15)( 3,25,13)( 4,26,14)( 5,31,19)( 6,32,20)( 7,29,17) ( 8,30,18)( 9,33,21)(10,34,22)(11,35,24)(12,36,23)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $72=2^{3} \cdot 3^{2}$ | 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: | 72.44 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 3A | 3B1 | 3B-1 | 3C1 | 3C-1 | 6A | 6B1 | 6B-1 | ||
Size | 1 | 3 | 3 | 9 | 2 | 4 | 4 | 8 | 8 | 6 | 12 | 12 | |
2 P | 1A | 1A | 1A | 1A | 3A | 3B-1 | 3B1 | 3C-1 | 3C1 | 3A | 3B1 | 3B-1 | |
3 P | 1A | 2A | 2B | 2C | 1A | 1A | 1A | 1A | 1A | 2A | 2B | 2B | |
Type | |||||||||||||
72.44.1a | R | ||||||||||||
72.44.1b | R | ||||||||||||
72.44.1c1 | C | ||||||||||||
72.44.1c2 | C | ||||||||||||
72.44.1d1 | C | ||||||||||||
72.44.1d2 | C | ||||||||||||
72.44.2a | R | ||||||||||||
72.44.2b1 | C | ||||||||||||
72.44.2b2 | C | ||||||||||||
72.44.3a | R | ||||||||||||
72.44.3b | R | ||||||||||||
72.44.6a | R |
magma: CharacterTable(G);