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Magma
magma: G := TransitiveGroup(45, 18);
Group action invariants
Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{15}:C_{12}$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,9,15,19,27,31,38,45,4,10,18,23,30,35,40)(2,7,14,21,26,33,37,44,6,11,17,22,29,34,41)(3,8,13,20,25,32,39,43,5,12,16,24,28,36,42), (1,43,38,25)(2,45,37,27)(3,44,39,26)(7,12,17,28)(8,10,16,30)(9,11,18,29)(13,24,42,32)(14,22,41,33)(15,23,40,31)(19,36)(20,34)(21,35) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $4$: $C_4$ $6$: $S_3$, $C_6$ $12$: $C_{12}$, $C_3 : C_4$ $18$: $S_3\times C_3$ $20$: $F_5$ $36$: $C_3\times (C_3 : C_4)$ $60$: $C_{15} : C_4$, $F_5\times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $F_5$
Degree 9: $S_3\times C_3$
Degree 15: $C_{15} : C_4$, $F_5\times C_3$
Low degree siblings
30T47Siblings 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 $ | $1$ | $1$ | $()$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1, 1, 1 $ | $15$ | $4$ | $( 4, 7,15,26)( 5, 9,13,27)( 6, 8,14,25)(10,19,38,30)(11,21,37,29)(12,20,39,28) (16,33)(17,31)(18,32)(22,43,41,36)(23,44,40,34)(24,45,42,35)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 4,15)( 5,13)( 6,14)( 7,26)( 8,25)( 9,27)(10,38)(11,37)(12,39)(19,30)(20,28) (21,29)(22,41)(23,40)(24,42)(34,44)(35,45)(36,43)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1, 1, 1 $ | $15$ | $4$ | $( 4,26,15, 7)( 5,27,13, 9)( 6,25,14, 8)(10,30,38,19)(11,29,37,21)(12,28,39,20) (16,33)(17,31)(18,32)(22,36,41,43)(23,34,40,44)(24,35,42,45)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,41,42) (43,45,44)$ | |
$ 12, 12, 12, 6, 3 $ | $15$ | $12$ | $( 1, 2, 3)( 4, 8,13,26, 6, 9,15,25, 5, 7,14,27)(10,21,39,30,11,20,38,29,12,19, 37,28)(16,32,17,33,18,31)(22,45,40,36,24,44,41,35,23,43,42,34)$ | |
$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $5$ | $6$ | $( 1, 2, 3)( 4,14, 5,15, 6,13)( 7,25, 9,26, 8,27)(10,37,12,38,11,39)(16,18,17) (19,29,20,30,21,28)(22,42,23,41,24,40)(31,33,32)(34,43,35,44,36,45)$ | |
$ 12, 12, 12, 6, 3 $ | $15$ | $12$ | $( 1, 2, 3)( 4,25,13, 7, 6,27,15, 8, 5,26,14, 9)(10,29,39,19,11,28,38,21,12,30, 37,20)(16,32,17,33,18,31)(22,35,40,43,24,34,41,45,23,36,42,44)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 5, 6)( 7, 9, 8)(10,12,11)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,38,39)(40,42,41) (43,44,45)$ | |
$ 12, 12, 12, 6, 3 $ | $15$ | $12$ | $( 1, 3, 2)( 4, 9,14,26, 5, 8,15,27, 6, 7,13,25)(10,20,37,30,12,21,38,28,11,19, 39,29)(16,31,18,33,17,32)(22,44,42,36,23,45,41,34,24,43,40,35)$ | |
$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $5$ | $6$ | $( 1, 3, 2)( 4,13, 6,15, 5,14)( 7,27, 8,26, 9,25)(10,39,11,38,12,37)(16,17,18) (19,28,21,30,20,29)(22,40,24,41,23,42)(31,32,33)(34,45,36,44,35,43)$ | |
$ 12, 12, 12, 6, 3 $ | $15$ | $12$ | $( 1, 3, 2)( 4,27,14, 7, 5,25,15, 9, 6,26,13, 8)(10,28,37,19,12,29,38,20,11,30, 39,21)(16,31,18,33,17,32)(22,34,42,43,23,35,41,44,24,36,40,45)$ | |
$ 15, 15, 15 $ | $4$ | $15$ | $( 1, 4, 9,10,15,18,19,23,27,30,31,35,38,40,45)( 2, 6, 7,11,14,17,21,22,26,29, 33,34,37,41,44)( 3, 5, 8,12,13,16,20,24,25,28,32,36,39,42,43)$ | |
$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $10$ | $6$ | $( 1, 4,18,19,31,35)( 2, 6,17,21,33,34)( 3, 5,16,20,32,36)( 7,29,22,44,37,14) ( 8,28,24,43,39,13)( 9,30,23,45,38,15)(10,40,27)(11,41,26)(12,42,25)$ | |
$ 15, 15, 15 $ | $4$ | $15$ | $( 1, 5, 7,10,13,17,19,24,26,30,32,34,38,42,44)( 2, 4, 8,11,15,16,21,23,25,29, 31,36,37,40,43)( 3, 6, 9,12,14,18,20,22,27,28,33,35,39,41,45)$ | |
$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $10$ | $6$ | $( 1, 5,17,19,32,34)( 2, 4,16,21,31,36)( 3, 6,18,20,33,35)( 7,30,24,44,38,13) ( 8,29,23,43,37,15)( 9,28,22,45,39,14)(10,42,26)(11,40,25)(12,41,27)$ | |
$ 15, 15, 15 $ | $4$ | $15$ | $( 1, 6, 8,10,14,16,19,22,25,30,33,36,38,41,43)( 2, 5, 9,11,13,18,21,24,27,29, 32,35,37,42,45)( 3, 4, 7,12,15,17,20,23,26,28,31,34,39,40,44)$ | |
$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $ | $10$ | $6$ | $( 1, 6,16,19,33,36)( 2, 5,18,21,32,35)( 3, 4,17,20,31,34)( 7,28,23,44,39,15) ( 8,30,22,43,38,14)( 9,29,24,45,37,13)(10,41,25)(11,42,27)(12,40,26)$ | |
$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1,10,19,30,38)( 2,11,21,29,37)( 3,12,20,28,39)( 4,15,23,31,40) ( 5,13,24,32,42)( 6,14,22,33,41)( 7,17,26,34,44)( 8,16,25,36,43) ( 9,18,27,35,45)$ | |
$ 15, 15, 15 $ | $4$ | $15$ | $( 1,11,20,30,37, 3,10,21,28,38, 2,12,19,29,39)( 4,14,24,31,41, 5,15,22,32,40, 6,13,23,33,42)( 7,16,27,34,43, 9,17,25,35,44, 8,18,26,36,45)$ | |
$ 15, 15, 15 $ | $4$ | $15$ | $( 1,12,21,30,39, 2,10,20,29,38, 3,11,19,28,37)( 4,13,22,31,42, 6,15,24,33,40, 5,14,23,32,41)( 7,18,25,34,45, 8,17,27,36,44, 9,16,26,35,43)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,16,33)( 2,18,32)( 3,17,31)( 4,20,34)( 5,21,35)( 6,19,36)( 7,23,39) ( 8,22,38)( 9,24,37)(10,25,41)(11,27,42)(12,26,40)(13,29,45)(14,30,43) (15,28,44)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,17,32)( 2,16,31)( 3,18,33)( 4,21,36)( 5,19,34)( 6,20,35)( 7,24,38) ( 8,23,37)( 9,22,39)(10,26,42)(11,25,40)(12,27,41)(13,30,44)(14,28,45) (15,29,43)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,18,31)( 2,17,33)( 3,16,32)( 4,19,35)( 5,20,36)( 6,21,34)( 7,22,37) ( 8,24,39)( 9,23,38)(10,27,40)(11,26,41)(12,25,42)(13,28,43)(14,29,44) (15,30,45)$ | |
$ 15, 15, 15 $ | $4$ | $15$ | $( 1,22,43,19,41,16,38,14,36,10,33, 8,30, 6,25)( 2,24,45,21,42,18,37,13,35,11, 32, 9,29, 5,27)( 3,23,44,20,40,17,39,15,34,12,31, 7,28, 4,26)$ | |
$ 15, 15, 15 $ | $4$ | $15$ | $( 1,23,45,19,40,18,38,15,35,10,31, 9,30, 4,27)( 2,22,44,21,41,17,37,14,34,11, 33, 7,29, 6,26)( 3,24,43,20,42,16,39,13,36,12,32, 8,28, 5,25)$ | |
$ 15, 15, 15 $ | $4$ | $15$ | $( 1,24,44,19,42,17,38,13,34,10,32, 7,30, 5,26)( 2,23,43,21,40,16,37,15,36,11, 31, 8,29, 4,25)( 3,22,45,20,41,18,39,14,35,12,33, 9,28, 6,27)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $180=2^{2} \cdot 3^{2} \cdot 5$ | 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: | 180.21 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 3A1 | 3A-1 | 3B | 3C1 | 3C-1 | 4A1 | 4A-1 | 5A | 6A1 | 6A-1 | 6B | 6C1 | 6C-1 | 12A1 | 12A-1 | 12A5 | 12A-5 | 15A1 | 15A-1 | 15B1 | 15B-1 | 15C1 | 15C-1 | 15C2 | 15C-2 | ||
Size | 1 | 5 | 1 | 1 | 2 | 2 | 2 | 15 | 15 | 4 | 5 | 5 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
2 P | 1A | 1A | 3A-1 | 3A1 | 3C1 | 3B | 3C-1 | 2A | 2A | 5A | 3A-1 | 3A1 | 3B | 3C1 | 3C-1 | 6A1 | 6A-1 | 6A-1 | 6A1 | 15C-1 | 15B1 | 15C-2 | 15A1 | 15C1 | 15B-1 | 15A-1 | 15C2 | |
3 P | 1A | 2A | 1A | 1A | 1A | 1A | 1A | 4A-1 | 4A1 | 5A | 2A | 2A | 2A | 2A | 2A | 4A1 | 4A-1 | 4A1 | 4A-1 | 5A | 5A | 5A | 5A | 5A | 5A | 5A | 5A | |
5 P | 1A | 2A | 3A-1 | 3A1 | 3C1 | 3B | 3C-1 | 4A1 | 4A-1 | 1A | 6A-1 | 6A1 | 6B | 6C-1 | 6C1 | 12A5 | 12A-5 | 12A1 | 12A-1 | 3C1 | 3B | 3C-1 | 3A-1 | 3C-1 | 3B | 3A1 | 3C1 | |
Type | ||||||||||||||||||||||||||||
180.21.1a | R | |||||||||||||||||||||||||||
180.21.1b | R | |||||||||||||||||||||||||||
180.21.1c1 | C | |||||||||||||||||||||||||||
180.21.1c2 | C | |||||||||||||||||||||||||||
180.21.1d1 | C | |||||||||||||||||||||||||||
180.21.1d2 | C | |||||||||||||||||||||||||||
180.21.1e1 | C | |||||||||||||||||||||||||||
180.21.1e2 | C | |||||||||||||||||||||||||||
180.21.1f1 | C | |||||||||||||||||||||||||||
180.21.1f2 | C | |||||||||||||||||||||||||||
180.21.1f3 | C | |||||||||||||||||||||||||||
180.21.1f4 | C | |||||||||||||||||||||||||||
180.21.2a | R | |||||||||||||||||||||||||||
180.21.2b | S | |||||||||||||||||||||||||||
180.21.2c1 | C | |||||||||||||||||||||||||||
180.21.2c2 | C | |||||||||||||||||||||||||||
180.21.2d1 | C | |||||||||||||||||||||||||||
180.21.2d2 | C | |||||||||||||||||||||||||||
180.21.4a | R | |||||||||||||||||||||||||||
180.21.4b1 | C | |||||||||||||||||||||||||||
180.21.4b2 | C | |||||||||||||||||||||||||||
180.21.4c1 | C | |||||||||||||||||||||||||||
180.21.4c2 | C | |||||||||||||||||||||||||||
180.21.4d1 | C | |||||||||||||||||||||||||||
180.21.4d2 | C | |||||||||||||||||||||||||||
180.21.4d3 | C | |||||||||||||||||||||||||||
180.21.4d4 | C |
magma: CharacterTable(G);