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Magma
magma: G := TransitiveGroup(45, 28);
Group action invariants
| Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $28$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_5^2:C_3^2$ | ||
| 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: | (1,43,23)(2,45,24)(3,44,22)(4,30,26)(5,28,25)(6,29,27)(7,31,11)(8,33,10)(9,32,12)(13,38,35)(14,39,34)(15,37,36)(16,40,21)(17,42,20)(18,41,19), (1,13,27)(2,14,26)(3,15,25)(4,18,39)(5,17,37)(6,16,38)(7,28,42)(8,30,41)(9,29,40)(10,22,36)(11,23,35)(12,24,34)(19,33,44)(20,31,43)(21,32,45) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ x 4 $9$: $C_3^2$ $75$: $C_5^2 : C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$ x 4
Degree 5: None
Degree 9: $C_3^2$
Degree 15: $C_5^2 : C_3$
Low degree siblings
45T28Siblings 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, 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 $ | $3$ | $15$ | $( 1, 3, 2)( 4,41,33,22,15, 5,42,31,23,13, 6,40,32,24,14)( 7,16,26,36,43, 9,18, 25,35,45, 8,17,27,34,44)(10,12,11)(19,21,20)(28,30,29)(37,39,38)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $25$ | $3$ | $( 1, 4, 7)( 2, 5, 9)( 3, 6, 8)(10,13,18)(11,14,17)(12,15,16)(19,23,26) (20,24,25)(21,22,27)(28,32,36)(29,33,35)(30,31,34)(37,40,44)(38,41,43) (39,42,45)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $25$ | $3$ | $( 1, 5, 8)( 2, 6, 7)( 3, 4, 9)(10,14,16)(11,15,18)(12,13,17)(19,24,27) (20,22,26)(21,23,25)(28,33,34)(29,31,36)(30,32,35)(37,41,45)(38,42,44) (39,40,43)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $25$ | $3$ | $( 1, 6, 9)( 2, 4, 8)( 3, 5, 7)(10,15,17)(11,13,16)(12,14,18)(19,22,25) (20,23,27)(21,24,26)(28,31,35)(29,32,34)(30,33,36)(37,42,43)(38,40,45) (39,41,44)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $25$ | $3$ | $( 1, 7, 4)( 2, 9, 5)( 3, 8, 6)(10,18,13)(11,17,14)(12,16,15)(19,26,23) (20,25,24)(21,27,22)(28,36,32)(29,35,33)(30,34,31)(37,44,40)(38,43,41) (39,45,42)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $25$ | $3$ | $( 1, 8, 5)( 2, 7, 6)( 3, 9, 4)(10,16,14)(11,18,15)(12,17,13)(19,27,24) (20,26,22)(21,25,23)(28,34,33)(29,36,31)(30,35,32)(37,45,41)(38,44,42) (39,43,40)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $25$ | $3$ | $( 1, 9, 6)( 2, 8, 4)( 3, 7, 5)(10,17,15)(11,16,13)(12,18,14)(19,25,22) (20,27,23)(21,26,24)(28,35,31)(29,34,32)(30,36,33)(37,43,42)(38,45,40) (39,44,41)$ | |
| $ 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: | 225.5 | magma: IdentifyGroup(G);
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| Character table: | 33 x 33 character table |
magma: CharacterTable(G);