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Magma
magma: G := TransitiveGroup(26, 17);
Group action invariants
Degree $n$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $17$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{13}^2:Q_8$ | ||
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: | (1,15,7,20)(2,18,6,17)(3,21,5,14)(4,24)(8,23,13,25)(9,26,12,22)(10,16,11,19), (1,11,13,3)(2,6,12,8)(4,9,10,5)(14,23,16,20)(17,25,26,18)(19,22,24,21) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $Q_8$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: None
Low degree siblings
26T17 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 | Representative |
$1^{26}$ | $1$ | $1$ | $()$ | |
$13^{2}$ | $8$ | $13$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$ | |
$13^{2}$ | $8$ | $13$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$ | |
$13^{2}$ | $8$ | $13$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$ | |
$13,1^{13}$ | $8$ | $13$ | $(14,17,20,23,26,16,19,22,25,15,18,21,24)$ | |
$13^{2}$ | $8$ | $13$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$ | |
$13^{2}$ | $8$ | $13$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$ | |
$13^{2}$ | $8$ | $13$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$ | |
$13^{2}$ | $8$ | $13$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$ | |
$13^{2}$ | $8$ | $13$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$ | |
$13^{2}$ | $8$ | $13$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$ | |
$13^{2}$ | $8$ | $13$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$ | |
$13^{2}$ | $8$ | $13$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$ | |
$13^{2}$ | $8$ | $13$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$ | |
$13,1^{13}$ | $8$ | $13$ | $(14,20,26,19,25,18,24,17,23,16,22,15,21)$ | |
$13^{2}$ | $8$ | $13$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$ | |
$13^{2}$ | $8$ | $13$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$ | |
$13^{2}$ | $8$ | $13$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$ | |
$13^{2}$ | $8$ | $13$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$ | |
$13^{2}$ | $8$ | $13$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$ | |
$13,1^{13}$ | $8$ | $13$ | $(14,26,25,24,23,22,21,20,19,18,17,16,15)$ | |
$13^{2}$ | $8$ | $13$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$ | |
$2^{12},1^{2}$ | $169$ | $2$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$ | |
$4^{6},2$ | $338$ | $4$ | $( 1,15, 7,20)( 2,18, 6,17)( 3,21, 5,14)( 4,24)( 8,23,13,25)( 9,26,12,22) (10,16,11,19)$ | |
$4^{6},1^{2}$ | $338$ | $4$ | $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,19,26,22)(16,24,25,17)(18,21,23,20)$ | |
$4^{6},2$ | $338$ | $4$ | $( 1,15,10,23)( 2,26, 9,25)( 3,24, 8,14)( 4,22, 7,16)( 5,20, 6,18)(11,21,13,17) (12,19)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $1352=2^{3} \cdot 13^{2}$ | 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: | 1352.44 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 4A | 4B | 4C | 13A1 | 13A2 | 13A4 | 13B1 | 13B2 | 13B4 | 13C1 | 13C2 | 13C4 | 13D1 | 13D2 | 13D3 | 13D4 | 13D5 | 13D6 | 13E1 | 13E2 | 13E3 | 13E4 | 13E5 | 13E6 | ||
Size | 1 | 169 | 338 | 338 | 338 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
2 P | 1A | 1A | 2A | 2A | 2A | 13A4 | 13D3 | 13E6 | 13D1 | 13A2 | 13B1 | 13C1 | 13E4 | 13B4 | 13C4 | 13E2 | 13A1 | 13D2 | 13B2 | 13C2 | 13E5 | 13D4 | 13D6 | 13E3 | 13D5 | 13E1 | |
13 P | 1A | 2A | 4C | 4A | 4B | 13A4 | 13D2 | 13E4 | 13D5 | 13A2 | 13B1 | 13C1 | 13E6 | 13B4 | 13C4 | 13E3 | 13A1 | 13D3 | 13B2 | 13C2 | 13E1 | 13D6 | 13D4 | 13E2 | 13D1 | 13E5 | |
Type | |||||||||||||||||||||||||||
1352.44.1a | R | ||||||||||||||||||||||||||
1352.44.1b | R | ||||||||||||||||||||||||||
1352.44.1c | R | ||||||||||||||||||||||||||
1352.44.1d | R | ||||||||||||||||||||||||||
1352.44.2a | S | ||||||||||||||||||||||||||
1352.44.8a1 | R | ||||||||||||||||||||||||||
1352.44.8a2 | R | ||||||||||||||||||||||||||
1352.44.8a3 | R | ||||||||||||||||||||||||||
1352.44.8b1 | R | ||||||||||||||||||||||||||
1352.44.8b2 | R | ||||||||||||||||||||||||||
1352.44.8b3 | R | ||||||||||||||||||||||||||
1352.44.8c1 | R | ||||||||||||||||||||||||||
1352.44.8c2 | R | ||||||||||||||||||||||||||
1352.44.8c3 | R | ||||||||||||||||||||||||||
1352.44.8d1 | R | ||||||||||||||||||||||||||
1352.44.8d2 | R | ||||||||||||||||||||||||||
1352.44.8d3 | R | ||||||||||||||||||||||||||
1352.44.8d4 | R | ||||||||||||||||||||||||||
1352.44.8d5 | R | ||||||||||||||||||||||||||
1352.44.8d6 | R | ||||||||||||||||||||||||||
1352.44.8e1 | R | ||||||||||||||||||||||||||
1352.44.8e2 | R | ||||||||||||||||||||||||||
1352.44.8e3 | R | ||||||||||||||||||||||||||
1352.44.8e4 | R | ||||||||||||||||||||||||||
1352.44.8e5 | R | ||||||||||||||||||||||||||
1352.44.8e6 | R |
magma: CharacterTable(G);