Show commands:
Magma
magma: G := TransitiveGroup(6, 13);
Group action invariants
Degree $n$: | $6$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $13$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3^2:D_4$ | ||
CHM label: | $F_{36}(6):2 = [S(3)^{2}]2 = S(3) wr 2$ | ||
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: | (2,4), (1,4)(2,5)(3,6), (2,4,6) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $D_{4}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Low degree siblings
6T13, 9T16, 12T34 x 2, 12T35 x 2, 12T36 x 2, 18T34 x 2, 18T36, 24T72 x 2, 36T53, 36T54 x 2Siblings 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 | Index | Representative |
1A | $1^{6}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{3}$ | $6$ | $2$ | $3$ | $(1,2)(3,4)(5,6)$ |
2B | $2,1^{4}$ | $6$ | $2$ | $1$ | $(3,5)$ |
2C | $2^{2},1^{2}$ | $9$ | $2$ | $2$ | $(3,5)(4,6)$ |
3A | $3^{2}$ | $4$ | $3$ | $4$ | $(1,5,3)(2,6,4)$ |
3B | $3,1^{3}$ | $4$ | $3$ | $2$ | $(1,5,3)$ |
4A | $4,2$ | $18$ | $4$ | $4$ | $(1,2)(3,4,5,6)$ |
6A | $6$ | $12$ | $6$ | $5$ | $(1,2,5,6,3,4)$ |
6B | $3,2,1$ | $12$ | $6$ | $3$ | $(2,6,4)(3,5)$ |
magma: ConjugacyClasses(G);
Malle's constant $a(G)$: $1$
Group invariants
Order: | $72=2^{3} \cdot 3^{2}$ | 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: | 72.40 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 3A | 3B | 4A | 6A | 6B | ||
Size | 1 | 6 | 6 | 9 | 4 | 4 | 18 | 12 | 12 | |
2 P | 1A | 1A | 1A | 1A | 3A | 3B | 2C | 3A | 3B | |
3 P | 1A | 2A | 2B | 2C | 1A | 1A | 4A | 2A | 2B | |
Type | ||||||||||
72.40.1a | R | |||||||||
72.40.1b | R | |||||||||
72.40.1c | R | |||||||||
72.40.1d | R | |||||||||
72.40.2a | R | |||||||||
72.40.4a | R | |||||||||
72.40.4b | R | |||||||||
72.40.4c | R | |||||||||
72.40.4d | R |
magma: CharacterTable(G);