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Magma
magma: G := TransitiveGroup(44, 11);
Group action invariants
| Degree $n$: | $44$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $11$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{11}\times A_4$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $11$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,13,28,39,6,19,30,42,12,24,33,4,16,25,38,7,18,31,43,9,21,36)(2,14,27,40,5,20,29,41,11,23,34,3,15,26,37,8,17,32,44,10,22,35), (1,11,19,28,34,42,6,15,24,30,37,4,12,17,25,33,44,7,16,22,31,38,2,9,18,27,36,43,5,13,21,29,39)(3,10,20,26,35,41,8,14,23,32,40) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ $11$: $C_{11}$ $12$: $A_4$ $33$: $C_{33}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: $A_4$
Degree 11: $C_{11}$
Degree 22: None
Low degree siblings
There are no siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $3$ | $( 2, 3, 4)( 5, 8, 7)( 9,11,10)(13,15,14)(17,20,19)(22,23,24)(25,27,26) (29,32,31)(34,35,36)(37,40,39)(41,42,44)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $3$ | $( 2, 4, 3)( 5, 7, 8)( 9,10,11)(13,14,15)(17,19,20)(22,24,23)(25,26,27) (29,31,32)(34,36,35)(37,39,40)(41,44,42)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)$ | |
| $ 22, 22 $ | $3$ | $22$ | $( 1, 5,12,15,18,22,28,29,33,37,43, 2, 6,11,16,17,21,27,30,34,38,44) ( 3, 7,10,13,20,24,26,31,35,39,41, 4, 8, 9,14,19,23,25,32,36,40,42)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1, 5, 9,16,17,24,28,29,36,38,44, 4, 6,11,13,18,22,25,30,34,39,43, 2, 7,12, 15,19,21,27,31,33,37,42)( 3, 8,10,14,20,23,26,32,35,40,41)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1, 5,10,16,17,23,28,29,35,38,44, 3, 6,11,14,18,22,26,30,34,40,43, 2, 8,12, 15,20,21,27,32,33,37,41)( 4, 7, 9,13,19,24,25,31,36,39,42)$ | |
| $ 11, 11, 11, 11 $ | $1$ | $11$ | $( 1, 6,12,16,18,21,28,30,33,38,43)( 2, 5,11,15,17,22,27,29,34,37,44) ( 3, 8,10,14,20,23,26,32,35,40,41)( 4, 7, 9,13,19,24,25,31,36,39,42)$ | |
| $ 22, 22 $ | $3$ | $22$ | $( 1, 9,18,25,33,42, 6,13,21,31,38, 4,12,19,28,36,43, 7,16,24,30,39) ( 2,10,17,26,34,41, 5,14,22,32,37, 3,11,20,27,35,44, 8,15,23,29,40)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1, 9,20,28,36,41, 6,13,23,30,39, 3,12,19,26,33,42, 8,16,24,32,38, 4,10,18, 25,35,43, 7,14,21,31,40)( 2,11,17,27,34,44, 5,15,22,29,37)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1, 9,17,28,36,44, 6,13,22,30,39, 2,12,19,27,33,42, 5,16,24,29,38, 4,11,18, 25,34,43, 7,15,21,31,37)( 3,10,20,26,35,41, 8,14,23,32,40)$ | |
| $ 11, 11, 11, 11 $ | $1$ | $11$ | $( 1,12,18,28,33,43, 6,16,21,30,38)( 2,11,17,27,34,44, 5,15,22,29,37) ( 3,10,20,26,35,41, 8,14,23,32,40)( 4, 9,19,25,36,42, 7,13,24,31,39)$ | |
| $ 22, 22 $ | $3$ | $22$ | $( 1,13,28,39, 6,19,30,42,12,24,33, 4,16,25,38, 7,18,31,43, 9,21,36) ( 2,14,27,40, 5,20,29,41,11,23,34, 3,15,26,37, 8,17,32,44,10,22,35)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1,13,26,38, 7,20,30,42,10,21,36, 3,16,25,40, 6,19,32,43, 9,23,33, 4,14,28, 39, 8,18,31,41,12,24,35)( 2,15,27,37, 5,17,29,44,11,22,34)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1,13,27,38, 7,17,30,42,11,21,36, 2,16,25,37, 6,19,29,43, 9,22,33, 4,15,28, 39, 5,18,31,44,12,24,34)( 3,14,26,40, 8,20,32,41,10,23,35)$ | |
| $ 11, 11, 11, 11 $ | $1$ | $11$ | $( 1,16,28,38, 6,18,30,43,12,21,33)( 2,15,27,37, 5,17,29,44,11,22,34) ( 3,14,26,40, 8,20,32,41,10,23,35)( 4,13,25,39, 7,19,31,42, 9,24,36)$ | |
| $ 22, 22 $ | $3$ | $22$ | $( 1,17,33, 5,21,37,12,27,43,15,30, 2,18,34, 6,22,38,11,28,44,16,29) ( 3,19,35, 7,23,39,10,25,41,13,32, 4,20,36, 8,24,40, 9,26,42,14,31)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1,17,36, 6,22,39,12,27,42,16,29, 4,18,34, 7,21,37, 9,28,44,13,30, 2,19,33, 5,24,38,11,25,43,15,31)( 3,20,35, 8,23,40,10,26,41,14,32)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1,17,35, 6,22,40,12,27,41,16,29, 3,18,34, 8,21,37,10,28,44,14,30, 2,20,33, 5,23,38,11,26,43,15,32)( 4,19,36, 7,24,39, 9,25,42,13,31)$ | |
| $ 11, 11, 11, 11 $ | $1$ | $11$ | $( 1,18,33, 6,21,38,12,28,43,16,30)( 2,17,34, 5,22,37,11,27,44,15,29) ( 3,20,35, 8,23,40,10,26,41,14,32)( 4,19,36, 7,24,39, 9,25,42,13,31)$ | |
| $ 11, 11, 11, 11 $ | $1$ | $11$ | $( 1,21,43,18,38,16,33,12,30, 6,28)( 2,22,44,17,37,15,34,11,29, 5,27) ( 3,23,41,20,40,14,35,10,32, 8,26)( 4,24,42,19,39,13,36, 9,31, 7,25)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1,21,43,18,38,16,33,12,30, 6,28)( 2,23,42,17,40,13,34,10,31, 5,26, 4,22,41, 19,37,14,36,11,32, 7,27, 3,24,44,20,39,15,35, 9,29, 8,25)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1,21,43,18,38,16,33,12,30, 6,28)( 2,24,41,17,39,14,34, 9,32, 5,25, 3,22,42, 20,37,13,35,11,31, 8,27, 4,23,44,19,40,15,36,10,29, 7,26)$ | |
| $ 22, 22 $ | $3$ | $22$ | $( 1,22,43,17,38,15,33,11,30, 5,28, 2,21,44,18,37,16,34,12,29, 6,27) ( 3,24,41,19,40,13,35, 9,32, 7,26, 4,23,42,20,39,14,36,10,31, 8,25)$ | |
| $ 22, 22 $ | $3$ | $22$ | $( 1,25, 6,31,12,36,16,39,18,42,21, 4,28, 7,30, 9,33,13,38,19,43,24) ( 2,26, 5,32,11,35,15,40,17,41,22, 3,27, 8,29,10,34,14,37,20,44,23)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1,25, 8,30, 9,35,16,39,20,43,24, 3,28, 7,32,12,36,14,38,19,41,21, 4,26, 6, 31,10,33,13,40,18,42,23)( 2,27, 5,29,11,34,15,37,17,44,22)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1,25, 5,30, 9,34,16,39,17,43,24, 2,28, 7,29,12,36,15,38,19,44,21, 4,27, 6, 31,11,33,13,37,18,42,22)( 3,26, 8,32,10,35,14,40,20,41,23)$ | |
| $ 11, 11, 11, 11 $ | $1$ | $11$ | $( 1,28, 6,30,12,33,16,38,18,43,21)( 2,27, 5,29,11,34,15,37,17,44,22) ( 3,26, 8,32,10,35,14,40,20,41,23)( 4,25, 7,31, 9,36,13,39,19,42,24)$ | |
| $ 22, 22 $ | $3$ | $22$ | $( 1,29,16,44,28,11,38,22, 6,34,18, 2,30,15,43,27,12,37,21, 5,33,17) ( 3,31,14,42,26, 9,40,24, 8,36,20, 4,32,13,41,25,10,39,23, 7,35,19)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1,29,13,43,27, 9,38,22, 7,33,17, 4,30,15,42,28,11,39,21, 5,36,18, 2,31,16, 44,25,12,37,24, 6,34,19)( 3,32,14,41,26,10,40,23, 8,35,20)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1,29,14,43,27,10,38,22, 8,33,17, 3,30,15,41,28,11,40,21, 5,35,18, 2,32,16, 44,26,12,37,23, 6,34,20)( 4,31,13,42,25, 9,39,24, 7,36,19)$ | |
| $ 11, 11, 11, 11 $ | $1$ | $11$ | $( 1,30,16,43,28,12,38,21, 6,33,18)( 2,29,15,44,27,11,37,22, 5,34,17) ( 3,32,14,41,26,10,40,23, 8,35,20)( 4,31,13,42,25, 9,39,24, 7,36,19)$ | |
| $ 11, 11, 11, 11 $ | $1$ | $11$ | $( 1,33,21,12,43,30,18, 6,38,28,16)( 2,34,22,11,44,29,17, 5,37,27,15) ( 3,35,23,10,41,32,20, 8,40,26,14)( 4,36,24, 9,42,31,19, 7,39,25,13)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1,33,21,12,43,30,18, 6,38,28,16)( 2,35,24,11,41,31,17, 8,39,27,14, 4,34,23, 9,44,32,19, 5,40,25,15, 3,36,22,10,42,29,20, 7,37,26,13)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1,33,21,12,43,30,18, 6,38,28,16)( 2,36,23,11,42,32,17, 7,40,27,13, 3,34,24, 10,44,31,20, 5,39,26,15, 4,35,22, 9,41,29,19, 8,37,25,14)$ | |
| $ 22, 22 $ | $3$ | $22$ | $( 1,34,21,11,43,29,18, 5,38,27,16, 2,33,22,12,44,30,17, 6,37,28,15) ( 3,36,23, 9,41,31,20, 7,40,25,14, 4,35,24,10,42,32,19, 8,39,26,13)$ | |
| $ 22, 22 $ | $3$ | $22$ | $( 1,37,30,22,16, 5,43,34,28,17,12, 2,38,29,21,15, 6,44,33,27,18,11) ( 3,39,32,24,14, 7,41,36,26,19,10, 4,40,31,23,13, 8,42,35,25,20, 9)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1,37,31,21,15, 7,43,34,25,18,11, 4,38,29,24,16, 5,42,33,27,19,12, 2,39,30, 22,13, 6,44,36,28,17, 9)( 3,40,32,23,14, 8,41,35,26,20,10)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1,37,32,21,15, 8,43,34,26,18,11, 3,38,29,23,16, 5,41,33,27,20,12, 2,40,30, 22,14, 6,44,35,28,17,10)( 4,39,31,24,13, 7,42,36,25,19, 9)$ | |
| $ 11, 11, 11, 11 $ | $1$ | $11$ | $( 1,38,30,21,16, 6,43,33,28,18,12)( 2,37,29,22,15, 5,44,34,27,17,11) ( 3,40,32,23,14, 8,41,35,26,20,10)( 4,39,31,24,13, 7,42,36,25,19, 9)$ | |
| $ 22, 22 $ | $3$ | $22$ | $( 1,41,38,35,30,26,21,20,16,10, 6, 3,43,40,33,32,28,23,18,14,12, 8) ( 2,42,37,36,29,25,22,19,15, 9, 5, 4,44,39,34,31,27,24,17,13,11, 7)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1,41,37,33,32,27,21,20,15,12, 8, 2,43,40,34,30,26,22,18,14,11, 6, 3,44,38, 35,29,28,23,17,16,10, 5)( 4,42,39,36,31,25,24,19,13, 9, 7)$ | |
| $ 33, 11 $ | $4$ | $33$ | $( 1,41,39,33,32,25,21,20,13,12, 8, 4,43,40,36,30,26,24,18,14, 9, 6, 3,42,38, 35,31,28,23,19,16,10, 7)( 2,44,37,34,29,27,22,17,15,11, 5)$ | |
| $ 11, 11, 11, 11 $ | $1$ | $11$ | $( 1,43,38,33,30,28,21,18,16,12, 6)( 2,44,37,34,29,27,22,17,15,11, 5) ( 3,41,40,35,32,26,23,20,14,10, 8)( 4,42,39,36,31,25,24,19,13, 9, 7)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $132=2^{2} \cdot 3 \cdot 11$ | 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: | 132.6 | magma: IdentifyGroup(G);
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| Character table: | 44 x 44 character table |
magma: CharacterTable(G);