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Magma
magma: G := TransitiveGroup(45, 29);
Group action invariants
| Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $29$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_5^2:C_9$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $15$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,42,16,3,41,17,2,40,18)(4,25,12,6,26,11,5,27,10)(7,39,32,8,38,31,9,37,33)(13,34,20,15,35,19,14,36,21)(22,43,30,24,44,29,23,45,28), (1,5,35,3,4,36,2,6,34)(7,19,24,8,21,23,9,20,22)(10,14,44,12,13,45,11,15,43)(16,28,33,17,30,32,18,29,31)(25,39,40,26,38,42,27,37,41) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ $9$: $C_9$ $75$: $C_5^2 : C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 5: None
Degree 9: $C_9$
Degree 15: $C_5^2 : C_3$
Low degree siblings
45T29Siblings 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, 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 $ | $1$ | $1$ | $()$ |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $5$ | $( 4,13,22,32,42)( 5,14,23,33,40)( 6,15,24,31,41)( 7,45,36,27,18) ( 8,43,34,25,16)( 9,44,35,26,17)$ |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $5$ | $( 4,22,42,13,32)( 5,23,40,14,33)( 6,24,41,15,31)( 7,36,18,45,27) ( 8,34,16,43,25)( 9,35,17,44,26)$ |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $5$ | $( 4,32,13,42,22)( 5,33,14,40,23)( 6,31,15,41,24)( 7,27,45,18,36) ( 8,25,43,16,34)( 9,26,44,17,35)$ |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $5$ | $( 4,42,32,22,13)( 5,40,33,23,14)( 6,41,31,24,15)( 7,18,27,36,45) ( 8,16,25,34,43)( 9,17,26,35,44)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 9, 8)(10,11,12)(13,14,15)(16,18,17)(19,20,21) (22,23,24)(25,27,26)(28,29,30)(31,32,33)(34,36,35)(37,38,39)(40,41,42) (43,45,44)$ |
| $ 15, 15, 3, 3, 3, 3, 3 $ | $3$ | $15$ | $( 1, 2, 3)( 4,14,24,32,40, 6,13,23,31,42, 5,15,22,33,41)( 7,44,34,27,17, 8,45, 35,25,18, 9,43,36,26,16)(10,11,12)(19,20,21)(28,29,30)(37,38,39)$ |
| $ 15, 15, 3, 3, 3, 3, 3 $ | $3$ | $15$ | $( 1, 2, 3)( 4,23,41,13,33, 6,22,40,15,32, 5,24,42,14,31)( 7,35,16,45,26, 8,36, 17,43,27, 9,34,18,44,25)(10,11,12)(19,20,21)(28,29,30)(37,38,39)$ |
| $ 15, 15, 3, 3, 3, 3, 3 $ | $3$ | $15$ | $( 1, 2, 3)( 4,33,15,42,23, 6,32,14,41,22, 5,31,13,40,24)( 7,26,43,18,35, 8,27, 44,16,36, 9,25,45,17,34)(10,11,12)(19,20,21)(28,29,30)(37,38,39)$ |
| $ 15, 15, 3, 3, 3, 3, 3 $ | $3$ | $15$ | $( 1, 2, 3)( 4,40,31,22,14, 6,42,33,24,13, 5,41,32,23,15)( 7,17,25,36,44, 8,18, 26,34,45, 9,16,27,35,43)(10,11,12)(19,20,21)(28,29,30)(37,38,39)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 8, 9)(10,12,11)(13,15,14)(16,17,18)(19,21,20) (22,24,23)(25,26,27)(28,30,29)(31,33,32)(34,35,36)(37,39,38)(40,42,41) (43,44,45)$ |
| $ 15, 15, 3, 3, 3, 3, 3 $ | $3$ | $15$ | $( 1, 3, 2)( 4,15,23,32,41, 5,13,24,33,42, 6,14,22,31,40)( 7,43,35,27,16, 9,45, 34,26,18, 8,44,36,25,17)(10,12,11)(19,21,20)(28,30,29)(37,39,38)$ |
| $ 15, 15, 3, 3, 3, 3, 3 $ | $3$ | $15$ | $( 1, 3, 2)( 4,24,40,13,31, 5,22,41,14,32, 6,23,42,15,33)( 7,34,17,45,25, 9,36, 16,44,27, 8,35,18,43,26)(10,12,11)(19,21,20)(28,30,29)(37,39,38)$ |
| $ 15, 15, 3, 3, 3, 3, 3 $ | $3$ | $15$ | $( 1, 3, 2)( 4,31,14,42,24, 5,32,15,40,22, 6,33,13,41,23)( 7,25,44,18,34, 9,27, 43,17,36, 8,26,45,16,35)(10,12,11)(19,21,20)(28,30,29)(37,39,38)$ |
| $ 15, 15, 3, 3, 3, 3, 3 $ | $3$ | $15$ | $( 1, 3, 2)( 4,41,33,22,15, 5,42,31,23,13, 6,40,32,24,14)( 7,16,26,36,43, 9,18, 25,35,45, 8,17,27,34,44)(10,12,11)(19,21,20)(28,30,29)(37,39,38)$ |
| $ 9, 9, 9, 9, 9 $ | $25$ | $9$ | $( 1, 4, 8, 3, 6, 9, 2, 5, 7)(10,13,16,12,15,17,11,14,18)(19,23,27,21,22,25,20, 24,26)(28,32,34,30,31,35,29,33,36)(37,40,45,39,42,43,38,41,44)$ |
| $ 9, 9, 9, 9, 9 $ | $25$ | $9$ | $( 1, 5, 9, 3, 4, 7, 2, 6, 8)(10,14,17,12,13,18,11,15,16)(19,24,25,21,23,26,20, 22,27)(28,33,35,30,32,36,29,31,34)(37,41,43,39,40,44,38,42,45)$ |
| $ 9, 9, 9, 9, 9 $ | $25$ | $9$ | $( 1, 6, 7, 3, 5, 8, 2, 4, 9)(10,15,18,12,14,16,11,13,17)(19,22,26,21,24,27,20, 23,25)(28,31,36,30,33,34,29,32,35)(37,42,44,39,41,45,38,40,43)$ |
| $ 9, 9, 9, 9, 9 $ | $25$ | $9$ | $( 1, 7, 5, 2, 9, 6, 3, 8, 4)(10,18,14,11,17,15,12,16,13)(19,26,24,20,25,22,21, 27,23)(28,36,33,29,35,31,30,34,32)(37,44,41,38,43,42,39,45,40)$ |
| $ 9, 9, 9, 9, 9 $ | $25$ | $9$ | $( 1, 8, 6, 2, 7, 4, 3, 9, 5)(10,16,15,11,18,13,12,17,14)(19,27,22,20,26,23,21, 25,24)(28,34,31,29,36,32,30,35,33)(37,45,42,38,44,40,39,43,41)$ |
| $ 9, 9, 9, 9, 9 $ | $25$ | $9$ | $( 1, 9, 4, 2, 8, 5, 3, 7, 6)(10,17,13,11,16,14,12,18,15)(19,25,23,20,27,24,21, 26,22)(28,35,32,29,34,33,30,36,31)(37,43,40,38,45,41,39,44,42)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $3$ | $5$ | $( 1,10,21,28,39)( 2,11,19,29,37)( 3,12,20,30,38)( 4,13,22,32,42) ( 5,14,23,33,40)( 6,15,24,31,41)( 7,36,18,45,27)( 8,34,16,43,25) ( 9,35,17,44,26)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $3$ | $5$ | $( 1,10,21,28,39)( 2,11,19,29,37)( 3,12,20,30,38)( 4,22,42,13,32) ( 5,23,40,14,33)( 6,24,41,15,31)( 7,27,45,18,36)( 8,25,43,16,34) ( 9,26,44,17,35)$ |
| $ 15, 15, 15 $ | $3$ | $15$ | $( 1,11,20,28,37, 3,10,19,30,39, 2,12,21,29,38)( 4,14,24,32,40, 6,13,23,31,42, 5,15,22,33,41)( 7,35,16,45,26, 8,36,17,43,27, 9,34,18,44,25)$ |
| $ 15, 15, 15 $ | $3$ | $15$ | $( 1,11,20,28,37, 3,10,19,30,39, 2,12,21,29,38)( 4,23,41,13,33, 6,22,40,15,32, 5,24,42,14,31)( 7,26,43,18,35, 8,27,44,16,36, 9,25,45,17,34)$ |
| $ 15, 15, 15 $ | $3$ | $15$ | $( 1,12,19,28,38, 2,10,20,29,39, 3,11,21,30,37)( 4,15,23,32,41, 5,13,24,33,42, 6,14,22,31,40)( 7,34,17,45,25, 9,36,16,44,27, 8,35,18,43,26)$ |
| $ 15, 15, 15 $ | $3$ | $15$ | $( 1,12,19,28,38, 2,10,20,29,39, 3,11,21,30,37)( 4,24,40,13,31, 5,22,41,14,32, 6,23,42,15,33)( 7,25,44,18,34, 9,27,43,17,36, 8,26,45,16,35)$ |
| $ 15, 15, 15 $ | $3$ | $15$ | $( 1,19,38,10,29, 3,21,37,12,28, 2,20,39,11,30)( 4,40,31,22,14, 6,42,33,24,13, 5,41,32,23,15)( 7,44,34,27,17, 8,45,35,25,18, 9,43,36,26,16)$ |
| $ 15, 15, 15 $ | $3$ | $15$ | $( 1,20,37,10,30, 2,21,38,11,28, 3,19,39,12,29)( 4,41,33,22,15, 5,42,31,23,13, 6,40,32,24,14)( 7,43,35,27,16, 9,45,34,26,18, 8,44,36,25,17)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $3$ | $5$ | $( 1,21,39,10,28)( 2,19,37,11,29)( 3,20,38,12,30)( 4,42,32,22,13) ( 5,40,33,23,14)( 6,41,31,24,15)( 7,45,36,27,18)( 8,43,34,25,16) ( 9,44,35,26,17)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $3$ | $5$ | $( 1,28,10,39,21)( 2,29,11,37,19)( 3,30,12,38,20)( 4,32,13,42,22) ( 5,33,14,40,23)( 6,31,15,41,24)( 7,45,36,27,18)( 8,43,34,25,16) ( 9,44,35,26,17)$ |
| $ 15, 15, 15 $ | $3$ | $15$ | $( 1,29,12,39,19, 3,28,11,38,21, 2,30,10,37,20)( 4,33,15,42,23, 6,32,14,41,22, 5,31,13,40,24)( 7,44,34,27,17, 8,45,35,25,18, 9,43,36,26,16)$ |
| $ 15, 15, 15 $ | $3$ | $15$ | $( 1,30,11,39,20, 2,28,12,37,21, 3,29,10,38,19)( 4,31,14,42,24, 5,32,15,40,22, 6,33,13,41,23)( 7,43,35,27,16, 9,45,34,26,18, 8,44,36,25,17)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $225=3^{2} \cdot 5^{2}$ | 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: | 225.3 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);