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Magma
magma: G := TransitiveGroup(45, 666);
Group action invariants
Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $666$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $\PSp(4,3)$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | yes | 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: | (1,3,6,12,19)(2,5,10,18,11)(4,8,15,22,29)(7,14,20,28,40)(9,16,24,33,13)(17,25,34,36,31)(21,30,41,45,32)(23,35,42,37,39)(26,27,38,44,43), (1,2,4,7,13)(3,5,9,14,8)(6,11,15,21,29)(10,17,24,31,19)(12,18,26,37,43)(16,23,34,35,33)(20,27,38,44,40)(22,32,41,45,30)(25,36,42,28,39) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 5: None
Degree 9: None
Degree 15: None
Low degree siblings
27T993, 36T12781, 40T14344, 40T14345Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $270$ | $2$ | $( 1, 6)( 2,10)( 4,15)( 7,20)(11,18)(12,19)(13,16)(17,26)(21,39)(22,29)(23,32) (24,33)(25,43)(27,31)(28,40)(30,37)(34,44)(35,45)(36,38)(41,42)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 1 $ | $3240$ | $4$ | $( 1,32, 6,23)( 2,40,10,28)( 4,42,15,41)( 5, 9)( 7,16,20,13)( 8,14) (11,45,18,35)(12,38,19,36)(17,34,26,44)(21,27,39,31)(22,30,29,37)(24,25,33,43)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $240$ | $3$ | $( 1,10,35)( 2,45, 6)( 3, 5, 9)( 4,36,44)( 7,37,21)(11,40,19)(12,18,28) (13,43,31)(15,38,34)(16,25,27)(17,32,41)(20,30,39)(23,42,26)$ |
$ 6, 6, 6, 6, 6, 6, 3, 2, 2, 1, 1 $ | $2160$ | $6$ | $( 1,45,10, 6,35, 2)( 3, 9, 5)( 4,34,36,15,44,38)( 7,39,37,20,21,30) (11,12,40,18,19,28)(13,27,43,16,31,25)(17,42,32,26,41,23)(22,29)(24,33)$ |
$ 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, 1, 1, 1, 1 $ | $45$ | $2$ | $( 1,32)( 2,42)( 4,40)( 6,23)(10,41)(11,44)(12,38)(13,16)(15,28)(17,35)(18,34) (19,36)(22,29)(25,43)(26,45)(27,31)$ |
$ 6, 6, 6, 6, 6, 3, 3, 3, 2, 1, 1, 1, 1 $ | $720$ | $6$ | $( 1,41,35,32,10,17)( 2,26, 6,42,45,23)( 3, 5, 9)( 4,19,44,40,36,11)( 7,37,21) (12,34,28,38,18,15)(13,25,31,16,43,27)(20,30,39)(22,29)$ |
$ 6, 6, 6, 6, 6, 3, 3, 3, 2, 1, 1, 1, 1 $ | $720$ | $6$ | $( 1,17,10,32,35,41)( 2,23,45,42, 6,26)( 3, 9, 5)( 4,11,36,40,44,19)( 7,21,37) (12,15,18,38,28,34)(13,27,43,16,31,25)(20,39,30)(22,29)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $40$ | $3$ | $( 1,40,37)( 2,42,14)( 3,11,41)( 4,16,24)( 6,29,26)( 8,18,44)( 9,10,34) (12,33,25)(13,32,21)(17,20,38)(27,28,36)(35,43,39)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $40$ | $3$ | $( 1,37,40)( 2,14,42)( 3,41,11)( 4,24,16)( 6,26,29)( 8,44,18)( 9,34,10) (12,25,33)(13,21,32)(17,38,20)(27,36,28)(35,39,43)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 1 $ | $540$ | $4$ | $( 1,27)( 2,17, 6, 8)( 3,25, 4,21)( 5,30,15,31)( 7,23,19,22)( 9,35)(10,43) (11,12,16,13)(14,38,26,44)(18,42,20,29)(24,32,41,33)(28,40)(34,39)(36,37)$ |
$ 6, 6, 6, 6, 3, 3, 3, 3, 2, 2, 2, 2, 1 $ | $360$ | $6$ | $( 1,37,40)( 2,26,42, 6,14,29)( 3,24,11, 4,41,16)( 5,15)( 7,19)( 8,38,18,17,44, 20)( 9,34,10)(12,21,33,13,25,32)(22,23)(27,36,28)(30,31)(35,39,43)$ |
$ 6, 6, 6, 6, 3, 3, 3, 3, 2, 2, 2, 2, 1 $ | $360$ | $6$ | $( 1,40,37)( 2,29,14, 6,42,26)( 3,16,41, 4,11,24)( 5,15)( 7,19)( 8,20,44,17,18, 38)( 9,10,34)(12,32,25,13,33,21)(22,23)(27,28,36)(30,31)(35,43,39)$ |
$ 12, 12, 6, 6, 4, 4, 1 $ | $2160$ | $12$ | $( 1,28,37,27,40,36)( 2,18,26,17,42,44, 6,20,14, 8,29,38)( 3,13,24,25,11,32, 4, 12,41,21,16,33)( 5,31,15,30)( 7,22,19,23)( 9,43,34,35,10,39)$ |
$ 12, 12, 6, 6, 4, 4, 1 $ | $2160$ | $12$ | $( 1,36,40,27,37,28)( 2,44,29,17,14,18, 6,38,42, 8,26,20)( 3,32,16,25,41,13, 4, 33,11,21,24,12)( 5,31,15,30)( 7,22,19,23)( 9,39,10,35,34,43)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $480$ | $3$ | $( 1,35,10)( 2,45, 6)( 3,30,37)( 4,28,16)( 5,39,21)( 7, 9,20)( 8,33,24) (11,34,31)(12,25,36)(13,40,15)(17,41,32)(18,27,44)(19,38,43)(23,42,26)$ |
$ 6, 6, 6, 6, 6, 3, 3, 3, 3, 2, 1 $ | $1440$ | $6$ | $( 1,41,35,32,10,17)( 2,23,45,42, 6,26)( 3,37,30)( 4,13,28,40,16,15)( 5,21,39) ( 7,20, 9)( 8,24,33)(11,27,34,44,31,18)(12,19,25,38,36,43)(22,29)$ |
$ 9, 9, 9, 9, 3, 3, 3 $ | $2880$ | $9$ | $( 1,13,29,40,32,26,37,21, 6)( 2, 8,33,42,18,25,14,44,12)( 3, 4,43,11,16,39,41, 24,35)( 5,22,30)( 7,31,19)( 9,36,17,10,27,20,34,28,38)(15,45,23)$ |
$ 9, 9, 9, 9, 3, 3, 3 $ | $2880$ | $9$ | $( 1,29,32,37, 6,13,40,26,21)( 2,33,18,14,12, 8,42,25,44)( 3,43,16,41,35, 4,11, 39,24)( 5,30,22)( 7,19,31)( 9,17,27,34,38,36,10,20,28)(15,23,45)$ |
$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $ | $5184$ | $5$ | $( 1,27,45,39,34)( 2,25,37,14, 5)( 3,11,35,26,24)( 4,22,36,33,44) ( 6,17,28,32,29)( 7, 8,10,30,41)( 9,18,38,19,21)(12,23,40,31,15) (13,16,43,20,42)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $25920=2^{6} \cdot 3^{4} \cdot 5$ | 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: | no | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 25920.a | magma: IdentifyGroup(G);
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Character table: |
2 6 6 2 2 2 5 3 2 3 3 3 3 1 1 . . 4 2 2 . 3 4 2 3 2 2 1 . 1 4 4 2 2 3 2 2 2 1 1 1 . 5 1 . . . . . . . . . . . . . . . . . . 1 1a 2a 3a 6a 6b 2b 4a 6c 3b 3c 6d 6e 3d 6f 9a 9b 4b 12a 12b 5a 2P 1a 1a 3a 3a 3a 1a 2b 3a 3c 3b 3b 3c 3d 3d 9b 9a 2a 6e 6d 5a 3P 1a 2a 1a 2a 2a 2b 4a 2b 1a 1a 2a 2a 1a 2a 3b 3c 4b 4b 4b 5a 5P 1a 2a 3a 6b 6a 2b 4a 6c 3c 3b 6e 6d 3d 6f 9b 9a 4b 12b 12a 1a 7P 1a 2a 3a 6a 6b 2b 4a 6c 3b 3c 6d 6e 3d 6f 9a 9b 4b 12a 12b 5a 11P 1a 2a 3a 6b 6a 2b 4a 6c 3c 3b 6e 6d 3d 6f 9b 9a 4b 12b 12a 5a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 5 -3 -1 A -A 1 -1 1 C /C H /H 2 . J /J 1 -/J -J . X.3 5 -3 -1 -A A 1 -1 1 /C C /H H 2 . /J J 1 -J -/J . X.4 6 -2 3 1 1 2 . -1 -3 -3 1 1 . -2 . . 2 -1 -1 1 X.5 10 2 1 -1 -1 -2 . 1 D /D C /C 1 -1 -J -/J 2 J /J . X.6 10 2 1 -1 -1 -2 . 1 /D D /C C 1 -1 -/J -J 2 /J J . X.7 15 -1 3 -1 -1 -1 -1 -1 6 6 2 2 . 2 . . 3 . . . X.8 15 7 . -2 -2 3 1 . -3 -3 1 1 3 1 . . -1 -1 -1 . X.9 20 4 5 1 1 4 . 1 2 2 -2 -2 -1 1 -1 -1 . . . . X.10 24 8 . 2 2 . . . 6 6 2 2 3 -1 . . . . . -1 X.11 30 -10 3 -1 -1 2 . -1 3 3 -1 -1 3 -1 . . -2 1 1 . X.12 30 6 -3 A -A 2 . -1 E /E H /H . . . . 2 /J J . X.13 30 6 -3 -A A 2 . -1 /E E /H H . . . . 2 J /J . X.14 40 -8 -2 B /B . . . F /F B /B 1 1 -J -/J . . . . X.15 40 -8 -2 /B B . . . /F F /B B 1 1 -/J -J . . . . X.16 45 -3 . . . -3 1 . G /G I /I . . . . 1 -/J -J . X.17 45 -3 . . . -3 1 . /G G /I I . . . . 1 -J -/J . X.18 60 -4 -3 -1 -1 4 . 1 6 6 2 2 -3 -1 . . . . . . X.19 64 . 4 . . . . . -8 -8 . . -2 . 1 1 . . . -1 X.20 81 9 . . . -3 -1 . . . . . . . . . -3 . . 1 A = E(3)-E(3)^2 = Sqrt(-3) = i3 B = -2*E(3)^2 = 1+Sqrt(-3) = 1+i3 C = -2*E(3)+E(3)^2 = (1-3*Sqrt(-3))/2 = -1-3b3 D = 5*E(3)+2*E(3)^2 = (-7+3*Sqrt(-3))/2 = -2+3b3 E = 6*E(3)-3*E(3)^2 = (-3+9*Sqrt(-3))/2 = 3+9b3 F = 2*E(3)+8*E(3)^2 = -5-3*Sqrt(-3) = -5-3i3 G = -9*E(3)^2 = (9+9*Sqrt(-3))/2 = 9+9b3 H = E(3)+2*E(3)^2 = (-3-Sqrt(-3))/2 = -2-b3 I = 3*E(3) = (-3+3*Sqrt(-3))/2 = 3b3 J = -E(3)^2 = (1+Sqrt(-3))/2 = 1+b3 |
magma: CharacterTable(G);