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Magma
magma: G := TransitiveGroup(44, 45);
Group action invariants
| Degree $n$: | $44$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{11}:C_5\times S_4$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,27,6,30,11,34,15,40,18,44,24,2,25,5,32,12,36,16,38,17,42,23,4,26,7,31,10,35,14,37,19,41,21,3,28,8,29,9,33,13,39,20,43,22), (1,10,5,30,18,4,9,8,29,19,3,12,7,32,20,2,11,6,31,17)(13,25,43,34,40,14,28,44,35,39,15,27,41,36,38,16,26,42,33,37)(21,24,23,22) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $5$: $C_5$ $6$: $S_3$ $10$: $C_{10}$ $24$: $S_4$ $30$: $S_3 \times C_5$ $55$: $C_{11}:C_5$ $110$: 22T5 $120$: 20T34 $330$: 33T7 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: $S_4$
Degree 11: $C_{11}:C_5$
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
| 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 $ | $1$ | $1$ | $()$ |
| $ 11, 11, 11, 11 $ | $5$ | $11$ | $( 1,11,18,25,36,42, 7,14,21,29,39)( 2,12,17,26,35,41, 8,13,22,30,40) ( 3, 9,20,27,34,44, 5,16,23,31,37)( 4,10,19,28,33,43, 6,15,24,32,38)$ |
| $ 11, 11, 11, 11 $ | $5$ | $11$ | $( 1,18,36, 7,21,39,11,25,42,14,29)( 2,17,35, 8,22,40,12,26,41,13,30) ( 3,20,34, 5,23,37, 9,27,44,16,31)( 4,19,33, 6,24,38,10,28,43,15,32)$ |
| $ 22, 22 $ | $15$ | $22$ | $( 1, 6,11,15,18,24,25,32,36,38,42, 4, 7,10,14,19,21,28,29,33,39,43) ( 2, 5,12,16,17,23,26,31,35,37,41, 3, 8, 9,13,20,22,27,30,34,40,44)$ |
| $ 22, 22 $ | $15$ | $22$ | $( 1,28, 7,32,11,33,14,38,18,43,21, 4,25, 6,29,10,36,15,39,19,42,24) ( 2,27, 8,31,12,34,13,37,17,44,22, 3,26, 5,30, 9,35,16,40,20,41,23)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39)(41,44)(42,43)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $3$ | $( 2, 4, 3)( 5, 8, 6)( 9,12,10)(13,15,16)(17,19,20)(22,24,23)(26,28,27) (30,32,31)(33,34,35)(37,40,38)(41,43,44)$ |
| $ 33, 11 $ | $40$ | $33$ | $( 1,11,18,25,36,42, 7,14,21,29,39)( 2,10,20,26,33,44, 8,15,23,30,38, 3,12,19, 27,35,43, 5,13,24,31,40, 4, 9,17,28,34,41, 6,16,22,32,37)$ |
| $ 33, 11 $ | $40$ | $33$ | $( 1,18,36, 7,21,39,11,25,42,14,29)( 2,19,34, 8,24,37,12,28,44,13,32, 3,17,33, 5,22,38, 9,26,43,16,30, 4,20,35, 6,23,40,10,27,41,15,31)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 3, 4)( 5, 6)( 9,10)(15,16)(19,20)(23,24)(27,28)(31,32)(33,34)(37,38)(43,44)$ |
| $ 22, 11, 11 $ | $30$ | $22$ | $( 1,11,18,25,36,42, 7,14,21,29,39)( 2,12,17,26,35,41, 8,13,22,30,40) ( 3,10,20,28,34,43, 5,15,23,32,37, 4, 9,19,27,33,44, 6,16,24,31,38)$ |
| $ 22, 11, 11 $ | $30$ | $22$ | $( 1,18,36, 7,21,39,11,25,42,14,29)( 2,17,35, 8,22,40,12,26,41,13,30) ( 3,19,34, 6,23,38, 9,28,44,15,31, 4,20,33, 5,24,37,10,27,43,16,32)$ |
| $ 44 $ | $30$ | $44$ | $( 1, 6,12,16,18,24,26,31,36,38,41, 3, 7,10,13,20,21,28,30,34,39,43, 2, 5,11, 15,17,23,25,32,35,37,42, 4, 8, 9,14,19,22,27,29,33,40,44)$ |
| $ 44 $ | $30$ | $44$ | $( 1,28, 8,31,11,33,13,37,18,43,22, 3,25, 6,30, 9,36,15,40,20,42,24, 2,27, 7, 32,12,34,14,38,17,44,21, 4,26, 5,29,10,35,16,39,19,41,23)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $6$ | $4$ | $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)(17,20,18,19)(21,24,22,23) (25,28,26,27)(29,32,30,31)(33,35,34,36)(37,39,38,40)(41,44,42,43)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ | $11$ | $5$ | $( 5,37,20,16,23)( 6,38,19,15,24)( 7,39,18,14,21)( 8,40,17,13,22) ( 9,31,34,27,44)(10,32,33,28,43)(11,29,36,25,42)(12,30,35,26,41)$ |
| $ 10, 10, 10, 10, 2, 2 $ | $33$ | $10$ | $( 1, 6,42,15,25, 4, 7,43,14,28)( 2, 5,41,16,26, 3, 8,44,13,27)( 9,35,31,40,23, 12,34,30,37,22)(10,36,32,39,24,11,33,29,38,21)(17,20)(18,19)$ |
| $ 15, 15, 5, 5, 3, 1 $ | $88$ | $15$ | $( 2, 4, 3)( 5,40,19,16,22, 6,37,17,15,23, 8,38,20,13,24)( 7,39,18,14,21) ( 9,30,33,27,41,10,31,35,28,44,12,32,34,26,43)(11,29,36,25,42)$ |
| $ 10, 10, 5, 5, 5, 5, 2, 1, 1 $ | $66$ | $10$ | $( 3, 4)( 5,38,20,15,23, 6,37,19,16,24)( 7,39,18,14,21)( 8,40,17,13,22) ( 9,32,34,28,44,10,31,33,27,43)(11,29,36,25,42)(12,30,35,26,41)$ |
| $ 20, 20, 4 $ | $66$ | $20$ | $( 1, 6,41,16,25, 4, 8,44,14,28, 2, 5,42,15,26, 3, 7,43,13,27)( 9,36,32,40,23, 11,33,30,37,21,10,35,31,39,24,12,34,29,38,22)(17,20,18,19)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ | $11$ | $5$ | $( 5,20,23,37,16)( 6,19,24,38,15)( 7,18,21,39,14)( 8,17,22,40,13) ( 9,34,44,31,27)(10,33,43,32,28)(11,36,42,29,25)(12,35,41,30,26)$ |
| $ 10, 10, 10, 10, 2, 2 $ | $33$ | $10$ | $( 1, 6,21,43,36, 4, 7,24,42,33)( 2, 5,22,44,35, 3, 8,23,41,34)( 9,40,20,26,16, 12,37,17,27,13)(10,39,19,25,15,11,38,18,28,14)(29,32)(30,31)$ |
| $ 15, 15, 5, 5, 3, 1 $ | $88$ | $15$ | $( 2, 4, 3)( 5,17,24,37,13, 6,20,22,38,16, 8,19,23,40,15)( 7,18,21,39,14) ( 9,35,43,31,26,10,34,41,32,27,12,33,44,30,28)(11,36,42,29,25)$ |
| $ 10, 10, 5, 5, 5, 5, 2, 1, 1 $ | $66$ | $10$ | $( 3, 4)( 5,19,23,38,16, 6,20,24,37,15)( 7,18,21,39,14)( 8,17,22,40,13) ( 9,33,44,32,27,10,34,43,31,28)(11,36,42,29,25)(12,35,41,30,26)$ |
| $ 20, 20, 4 $ | $66$ | $20$ | $( 1, 6,22,44,36, 4, 8,23,42,33, 2, 5,21,43,35, 3, 7,24,41,34)( 9,39,19,26,16, 11,38,17,27,14,10,40,20,25,15,12,37,18,28,13)(29,32,30,31)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ | $11$ | $5$ | $( 5,23,16,20,37)( 6,24,15,19,38)( 7,21,14,18,39)( 8,22,13,17,40) ( 9,44,27,34,31)(10,43,28,33,32)(11,42,25,36,29)(12,41,26,35,30)$ |
| $ 10, 10, 10, 10, 2, 2 $ | $33$ | $10$ | $( 1, 6,25,38,11, 4, 7,28,39,10)( 2, 5,26,37,12, 3, 8,27,40, 9)(13,23,17,44,30, 16,22,20,41,31)(14,24,18,43,29,15,21,19,42,32)(33,36)(34,35)$ |
| $ 15, 15, 5, 5, 3, 1 $ | $88$ | $15$ | $( 2, 4, 3)( 5,22,15,20,40, 6,23,13,19,37, 8,24,16,17,38)( 7,21,14,18,39) ( 9,41,28,34,30,10,44,26,33,31,12,43,27,35,32)(11,42,25,36,29)$ |
| $ 10, 10, 5, 5, 5, 5, 2, 1, 1 $ | $66$ | $10$ | $( 3, 4)( 5,24,16,19,37, 6,23,15,20,38)( 7,21,14,18,39)( 8,22,13,17,40) ( 9,43,27,33,31,10,44,28,34,32)(11,42,25,36,29)(12,41,26,35,30)$ |
| $ 20, 20, 4 $ | $66$ | $20$ | $( 1, 6,26,37,11, 4, 8,27,39,10, 2, 5,25,38,12, 3, 7,28,40, 9)(13,23,18,43,30, 16,21,19,41,31,14,24,17,44,29,15,22,20,42,32)(33,35,34,36)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ | $11$ | $5$ | $( 5,16,37,23,20)( 6,15,38,24,19)( 7,14,39,21,18)( 8,13,40,22,17) ( 9,27,31,44,34)(10,28,32,43,33)(11,25,29,42,36)(12,26,30,41,35)$ |
| $ 10, 10, 10, 10, 2, 2 $ | $33$ | $10$ | $( 1, 6,18,10,29, 4, 7,19,11,32)( 2, 5,17, 9,30, 3, 8,20,12,31)(13,44,40,27,35, 16,41,37,26,34)(14,43,39,28,36,15,42,38,25,33)(21,24)(22,23)$ |
| $ 15, 15, 5, 5, 3, 1 $ | $88$ | $15$ | $( 2, 4, 3)( 5,13,38,23,17, 6,16,40,24,20, 8,15,37,22,19)( 7,14,39,21,18) ( 9,26,32,44,35,10,27,30,43,34,12,28,31,41,33)(11,25,29,42,36)$ |
| $ 10, 10, 5, 5, 5, 5, 2, 1, 1 $ | $66$ | $10$ | $( 3, 4)( 5,15,37,24,20, 6,16,38,23,19)( 7,14,39,21,18)( 8,13,40,22,17) ( 9,28,31,43,34,10,27,32,44,33)(11,25,29,42,36)(12,26,30,41,35)$ |
| $ 20, 20, 4 $ | $66$ | $20$ | $( 1, 6,17, 9,29, 4, 8,20,11,32, 2, 5,18,10,30, 3, 7,19,12,31)(13,44,39,28,35, 16,42,38,26,34,14,43,40,27,36,15,41,37,25,33)(21,24,22,23)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $1320=2^{3} \cdot 3 \cdot 5 \cdot 11$ | 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: | 1320.139 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);