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Magma
magma: G := TransitiveGroup(27, 35);
Group action invariants
Degree $n$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3:S_3^2$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,19,3,21,2,20)(4,16,6,18,5,17)(7,14,9,13,8,15)(10,11,12)(22,25,24,27,23,26), (1,26,4)(2,25,5,3,27,6)(7,23,10,17,13,20)(8,22,11,16,14,19)(9,24,12,18,15,21) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ x 3 $12$: $D_{6}$ x 3 $36$: $S_3^2$ x 3 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$ x 3
Degree 9: $S_3^2$ x 3
Low degree siblings
12T71, 18T53 x 3, 36T88 x 3, 36T93Siblings 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 $ | $1$ | $1$ | $()$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $9$ | $2$ | $( 4,26)( 5,27)( 6,25)( 7,22)( 8,23)( 9,24)(10,19)(11,20)(12,21)(13,16)(14,17) (15,18)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $9$ | $2$ | $( 2, 3)( 5, 6)( 7,17)( 8,16)( 9,18)(10,20)(11,19)(12,21)(13,23)(14,22)(15,24) (25,27)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $9$ | $2$ | $( 2, 3)( 4,26)( 5,25)( 6,27)( 7,14)( 8,13)( 9,15)(10,11)(16,23)(17,22)(18,24) (19,20)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$ | |
$ 6, 6, 6, 6, 3 $ | $18$ | $6$ | $( 1, 2, 3)( 4,27, 6,26, 5,25)( 7,23, 9,22, 8,24)(10,20,12,19,11,21) (13,17,15,16,14,18)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 4,26)( 2, 5,27)( 3, 6,25)( 7,10,13)( 8,11,14)( 9,12,15)(16,19,22) (17,20,23)(18,21,24)$ | |
$ 6, 6, 6, 6, 3 $ | $18$ | $6$ | $( 1, 4,26)( 2, 6,27, 3, 5,25)( 7,20,13,17,10,23)( 8,19,14,16,11,22) ( 9,21,15,18,12,24)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 5,25)( 2, 6,26)( 3, 4,27)( 7,11,15)( 8,12,13)( 9,10,14)(16,20,24) (17,21,22)(18,19,23)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 7,23)( 2, 8,24)( 3, 9,22)( 4,10,17)( 5,11,18)( 6,12,16)(13,20,26) (14,21,27)(15,19,25)$ | |
$ 6, 6, 6, 6, 3 $ | $18$ | $6$ | $( 1, 7,21,27,12,16)( 2, 9,19,26,10,18)( 3, 8,20,25,11,17)( 4,13,24, 5,15,22) ( 6,14,23)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 8,22)( 2, 9,23)( 3, 7,24)( 4,11,16)( 5,12,17)( 6,10,18)(13,21,25) (14,19,26)(15,20,27)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 9,24)( 2, 7,22)( 3, 8,23)( 4,12,18)( 5,10,16)( 6,11,17)(13,19,27) (14,20,25)(15,21,26)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,10,20)( 2,11,21)( 3,12,19)( 4,13,23)( 5,14,24)( 6,15,22)( 7,17,26) ( 8,18,27)( 9,16,25)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,12,21)( 2,10,19)( 3,11,20)( 4,15,24)( 5,13,22)( 6,14,23)( 7,16,27) ( 8,17,25)( 9,18,26)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $108=2^{2} \cdot 3^{3}$ | 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: | 108.40 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 3G1 | 3G-1 | 6A | 6B | 6C | ||
Size | 1 | 9 | 9 | 9 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 18 | 18 | 18 | |
2 P | 1A | 1A | 1A | 1A | 3B | 3C | 3A | 3F | 3D | 3G1 | 3E | 3G-1 | 3A | 3B | 3C | |
3 P | 1A | 2A | 2B | 2C | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2A | 2B | 2C | |
Type | ||||||||||||||||
108.40.1a | R | |||||||||||||||
108.40.1b | R | |||||||||||||||
108.40.1c | R | |||||||||||||||
108.40.1d | R | |||||||||||||||
108.40.2a | R | |||||||||||||||
108.40.2b | R | |||||||||||||||
108.40.2c | R | |||||||||||||||
108.40.2d | R | |||||||||||||||
108.40.2e | R | |||||||||||||||
108.40.2f | R | |||||||||||||||
108.40.4a | R | |||||||||||||||
108.40.4b | R | |||||||||||||||
108.40.4c | R | |||||||||||||||
108.40.4d1 | C | |||||||||||||||
108.40.4d2 | C |
magma: CharacterTable(G);