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Magma
magma: G := TransitiveGroup(37, 5);
Group action invariants
Degree $n$: | $37$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $5$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{37}:C_{6}$ | ||
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: | (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37), (1,27,26,36,10,11)(2,17,15,35,20,22)(3,7,4,34,30,33)(5,24,19,32,13,18)(6,14,8,31,23,29)(9,21,12,28,16,25) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ Resolvents shown for degrees $\leq 47$
Subfields
Prime degree - 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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $37$ | $3$ | $( 2,11,27)( 3,21,16)( 4,31, 5)( 6,14,20)( 7,24, 9)( 8,34,35)(10,17,13) (12,37,28)(15,30,32)(18,23,36)(19,33,25)(22,26,29)$ |
$ 6, 6, 6, 6, 6, 6, 1 $ | $37$ | $6$ | $( 2,12,11,37,27,28)( 3,23,21,36,16,18)( 4,34,31,35, 5, 8)( 6,19,14,33,20,25) ( 7,30,24,32, 9,15)(10,26,17,29,13,22)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $37$ | $3$ | $( 2,27,11)( 3,16,21)( 4, 5,31)( 6,20,14)( 7, 9,24)( 8,35,34)(10,13,17) (12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26)$ |
$ 6, 6, 6, 6, 6, 6, 1 $ | $37$ | $6$ | $( 2,28,27,37,11,12)( 3,18,16,36,21,23)( 4, 8, 5,35,31,34)( 6,25,20,33,14,19) ( 7,15, 9,32,24,30)(10,22,13,29,17,26)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $37$ | $2$ | $( 2,37)( 3,36)( 4,35)( 5,34)( 6,33)( 7,32)( 8,31)( 9,30)(10,29)(11,28)(12,27) (13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)$ |
$ 37 $ | $6$ | $37$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37)$ |
$ 37 $ | $6$ | $37$ | $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37, 2, 4, 6, 8,10,12, 14,16,18,20,22,24,26,28,30,32,34,36)$ |
$ 37 $ | $6$ | $37$ | $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37, 3, 6, 9,12,15,18,21,24,27,30,33,36, 2, 5, 8,11,14,17,20,23,26,29,32,35)$ |
$ 37 $ | $6$ | $37$ | $( 1, 6,11,16,21,26,31,36, 4, 9,14,19,24,29,34, 2, 7,12,17,22,27,32,37, 5,10, 15,20,25,30,35, 3, 8,13,18,23,28,33)$ |
$ 37 $ | $6$ | $37$ | $( 1, 7,13,19,25,31,37, 6,12,18,24,30,36, 5,11,17,23,29,35, 4,10,16,22,28,34, 3, 9,15,21,27,33, 2, 8,14,20,26,32)$ |
$ 37 $ | $6$ | $37$ | $( 1,10,19,28,37, 9,18,27,36, 8,17,26,35, 7,16,25,34, 6,15,24,33, 5,14,23,32, 4,13,22,31, 3,12,21,30, 2,11,20,29)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $222=2 \cdot 3 \cdot 37$ | 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: | 222.1 | magma: IdentifyGroup(G);
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Character table: |
2 1 1 1 1 1 1 . . . . . . 3 1 1 1 1 1 1 . . . . . . 37 1 . . . . . 1 1 1 1 1 1 1a 3a 6a 3b 6b 2a 37a 37b 37c 37d 37e 37f 2P 1a 3b 3a 3a 3b 1a 37b 37c 37e 37a 37f 37d 3P 1a 1a 2a 1a 2a 2a 37c 37e 37f 37b 37d 37a 5P 1a 3b 6b 3a 6a 2a 37d 37a 37b 37f 37c 37e 7P 1a 3a 6a 3b 6b 2a 37c 37e 37f 37b 37d 37a 11P 1a 3b 6b 3a 6a 2a 37a 37b 37c 37d 37e 37f 13P 1a 3a 6a 3b 6b 2a 37d 37a 37b 37f 37c 37e 17P 1a 3b 6b 3a 6a 2a 37b 37c 37e 37a 37f 37d 19P 1a 3a 6a 3b 6b 2a 37d 37a 37b 37f 37c 37e 23P 1a 3b 6b 3a 6a 2a 37e 37f 37d 37c 37a 37b 29P 1a 3b 6b 3a 6a 2a 37e 37f 37d 37c 37a 37b 31P 1a 3a 6a 3b 6b 2a 37e 37f 37d 37c 37a 37b 37P 1a 3a 6a 3b 6b 2a 1a 1a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 1 -1 -1 1 1 1 1 1 1 X.3 1 A -/A /A -A -1 1 1 1 1 1 1 X.4 1 /A -A A -/A -1 1 1 1 1 1 1 X.5 1 A /A /A A 1 1 1 1 1 1 1 X.6 1 /A A A /A 1 1 1 1 1 1 1 X.7 6 . . . . . B G F C E D X.8 6 . . . . . C B G D F E X.9 6 . . . . . D C B E G F X.10 6 . . . . . E D C F B G X.11 6 . . . . . F E D G C B X.12 6 . . . . . G F E B D C A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(37)^3+E(37)^4+E(37)^7+E(37)^30+E(37)^33+E(37)^34 C = E(37)^2+E(37)^15+E(37)^17+E(37)^20+E(37)^22+E(37)^35 D = E(37)+E(37)^10+E(37)^11+E(37)^26+E(37)^27+E(37)^36 E = E(37)^5+E(37)^13+E(37)^18+E(37)^19+E(37)^24+E(37)^32 F = E(37)^9+E(37)^12+E(37)^16+E(37)^21+E(37)^25+E(37)^28 G = E(37)^6+E(37)^8+E(37)^14+E(37)^23+E(37)^29+E(37)^31 |
magma: CharacterTable(G);