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Magma
magma: G := TransitiveGroup(26, 30);
Group invariants
Abstract group: | $C_{13}^2:(C_4\times S_3)$ | magma: IdentifyGroup(G);
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Order: | $4056=2^{3} \cdot 3 \cdot 13^{2}$ | 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 | magma: NilpotencyClass(G);
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Group action invariants
Degree $n$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $30$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | $(1,17,5,24)(2,22,4,19)(3,14)(6,16,13,25)(7,21,12,20)(8,26,11,15)(9,18,10,23)$, $(1,17,6,24,11,18,3,25,8,19,13,26,5,20,10,14,2,21,7,15,12,22,4,16,9,23)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $6$: $S_3$ $8$: $C_4\times C_2$ $12$: $D_{6}$ $24$: $S_3 \times C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: None
Low degree siblings
39T42 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 | $1^{26}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{13}$ | $39$ | $2$ | $13$ | $( 1,23)( 2,22)( 3,21)( 4,20)( 5,19)( 6,18)( 7,17)( 8,16)( 9,15)(10,14)(11,26)(12,25)(13,24)$ |
2B | $2^{13}$ | $39$ | $2$ | $13$ | $( 1,26)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7,19)( 8,20)( 9,21)(10,22)(11,23)(12,24)(13,25)$ |
2C | $2^{12},1^{2}$ | $169$ | $2$ | $12$ | $( 1, 7)( 2, 6)( 3, 5)( 8,13)( 9,12)(10,11)(14,20)(15,19)(16,18)(21,26)(22,25)(23,24)$ |
3A | $3^{8},1^{2}$ | $338$ | $3$ | $16$ | $( 1, 8, 3)( 2,11,12)( 5, 7,13)( 6,10, 9)(14,16,21)(15,25,24)(18,26,20)(19,22,23)$ |
4A1 | $4^{6},1^{2}$ | $169$ | $4$ | $18$ | $( 1, 2, 7, 6)( 3,12, 5, 9)( 8,11,13,10)(14,15,20,19)(16,25,18,22)(21,24,26,23)$ |
4A-1 | $4^{6},1^{2}$ | $169$ | $4$ | $18$ | $( 1, 6, 7, 2)( 3, 9, 5,12)( 8,10,13,11)(14,19,20,15)(16,22,18,25)(21,23,26,24)$ |
4B1 | $4^{6},2$ | $507$ | $4$ | $19$ | $( 1,22,10,20)( 2,16, 9,26)( 3,23, 8,19)( 4,17, 7,25)( 5,24, 6,18)(11,14,13,15)(12,21)$ |
4B-1 | $4^{6},2$ | $507$ | $4$ | $19$ | $( 1,20,10,22)( 2,26, 9,16)( 3,19, 8,23)( 4,25, 7,17)( 5,18, 6,24)(11,15,13,14)(12,21)$ |
6A | $6^{4},1^{2}$ | $338$ | $6$ | $20$ | $( 1, 5, 8, 7, 3,13)( 2, 9,11, 6,12,10)(14,26,16,20,21,18)(15,23,25,19,24,22)$ |
12A1 | $12^{2},1^{2}$ | $338$ | $12$ | $22$ | $( 1,10, 5, 2, 8, 9, 7,11, 3, 6,13,12)(14,22,26,15,16,23,20,25,21,19,18,24)$ |
12A-1 | $12^{2},1^{2}$ | $338$ | $12$ | $22$ | $( 1,12,13, 6, 3,11, 7, 9, 8, 2, 5,10)(14,24,18,19,21,25,20,23,16,15,26,22)$ |
13A1 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,26,25,24,23,22,21,20,19,18,17,16,15)$ |
13A2 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,25,23,21,19,17,15,26,24,22,20,18,16)$ |
13A4 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,23,19,15,24,20,16,25,21,17,26,22,18)$ |
13B1 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,20,26,19,25,18,24,17,23,16,22,15,21)$ |
13B2 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,26,25,24,23,22,21,20,19,18,17,16,15)$ |
13B4 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,25,23,21,19,17,15,26,24,22,20,18,16)$ |
13C | $13,1^{13}$ | $24$ | $13$ | $12$ | $(14,18,22,26,17,21,25,16,20,24,15,19,23)$ |
13D1 | $13^{2}$ | $24$ | $13$ | $24$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,21,15,22,16,23,17,24,18,25,19,26,20)$ |
13D2 | $13^{2}$ | $24$ | $13$ | $24$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,25,23,21,19,17,15,26,24,22,20,18,16)$ |
13D4 | $13^{2}$ | $24$ | $13$ | $24$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,26,25,24,23,22,21,20,19,18,17,16,15)$ |
26A1 | $26$ | $156$ | $26$ | $25$ | $( 1,16, 2,15, 3,14, 4,26, 5,25, 6,24, 7,23, 8,22, 9,21,10,20,11,19,12,18,13,17)$ |
26A3 | $26$ | $156$ | $26$ | $25$ | $( 1,15, 4,25, 7,22,10,19,13,16, 3,26, 6,23, 9,20,12,17, 2,14, 5,24, 8,21,11,18)$ |
26A7 | $26$ | $156$ | $26$ | $25$ | $( 1,26, 8,19, 2,25, 9,18, 3,24,10,17, 4,23,11,16, 5,22,12,15, 6,21,13,14, 7,20)$ |
26B1 | $26$ | $156$ | $26$ | $25$ | $( 1,16, 7,22,13,15, 6,21,12,14, 5,20,11,26, 4,19,10,25, 3,18, 9,24, 2,17, 8,23)$ |
26B3 | $26$ | $156$ | $26$ | $25$ | $( 1,22, 6,14,11,19, 3,24, 8,16,13,21, 5,26,10,18, 2,23, 7,15,12,20, 4,25, 9,17)$ |
26B7 | $26$ | $156$ | $26$ | $25$ | $( 1,21, 4,24, 7,14,10,17,13,20, 3,23, 6,26, 9,16,12,19, 2,22, 5,25, 8,15,11,18)$ |
Malle's constant $a(G)$: $1/12$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 2B | 2C | 3A | 4A1 | 4A-1 | 4B1 | 4B-1 | 6A | 12A1 | 12A-1 | 13A1 | 13A2 | 13A4 | 13B1 | 13B2 | 13B4 | 13C | 13D1 | 13D2 | 13D4 | 26A1 | 26A3 | 26A7 | 26B1 | 26B3 | 26B7 | ||
Size | 1 | 39 | 39 | 169 | 338 | 169 | 169 | 507 | 507 | 338 | 338 | 338 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 156 | 156 | 156 | 156 | 156 | 156 | |
2 P | 1A | 1A | 1A | 1A | 3A | 2C | 2C | 2C | 2C | 3A | 6A | 6A | 13A2 | 13A4 | 13A1 | 13B2 | 13B4 | 13B1 | 13C | 13D2 | 13D4 | 13D1 | 13A1 | 13A2 | 13A4 | 13B1 | 13B2 | 13B4 | |
3 P | 1A | 2A | 2B | 2C | 1A | 4A-1 | 4A1 | 4B-1 | 4B1 | 2C | 4A1 | 4A-1 | 13A2 | 13A4 | 13A1 | 13B2 | 13B4 | 13B1 | 13C | 13D2 | 13D4 | 13D1 | 26A3 | 26A7 | 26A1 | 26B3 | 26B7 | 26B1 | |
13 P | 1A | 2A | 2B | 2C | 3A | 4A1 | 4A-1 | 4B1 | 4B-1 | 6A | 12A1 | 12A-1 | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2B | 2B | 2B | |
Type | |||||||||||||||||||||||||||||
4056.bb.1a | R | ||||||||||||||||||||||||||||
4056.bb.1b | R | ||||||||||||||||||||||||||||
4056.bb.1c | R | ||||||||||||||||||||||||||||
4056.bb.1d | R | ||||||||||||||||||||||||||||
4056.bb.1e1 | C | ||||||||||||||||||||||||||||
4056.bb.1e2 | C | ||||||||||||||||||||||||||||
4056.bb.1f1 | C | ||||||||||||||||||||||||||||
4056.bb.1f2 | C | ||||||||||||||||||||||||||||
4056.bb.2a | R | ||||||||||||||||||||||||||||
4056.bb.2b | R | ||||||||||||||||||||||||||||
4056.bb.2c1 | C | ||||||||||||||||||||||||||||
4056.bb.2c2 | C | ||||||||||||||||||||||||||||
4056.bb.12a1 | R | ||||||||||||||||||||||||||||
4056.bb.12a2 | R | ||||||||||||||||||||||||||||
4056.bb.12a3 | R | ||||||||||||||||||||||||||||
4056.bb.12b1 | R | ||||||||||||||||||||||||||||
4056.bb.12b2 | R | ||||||||||||||||||||||||||||
4056.bb.12b3 | R | ||||||||||||||||||||||||||||
4056.bb.12c1 | R | ||||||||||||||||||||||||||||
4056.bb.12c2 | R | ||||||||||||||||||||||||||||
4056.bb.12c3 | R | ||||||||||||||||||||||||||||
4056.bb.12d1 | R | ||||||||||||||||||||||||||||
4056.bb.12d2 | R | ||||||||||||||||||||||||||||
4056.bb.12d3 | R | ||||||||||||||||||||||||||||
4056.bb.24a | R | ||||||||||||||||||||||||||||
4056.bb.24b1 | R | ||||||||||||||||||||||||||||
4056.bb.24b2 | R | ||||||||||||||||||||||||||||
4056.bb.24b3 | R |
magma: CharacterTable(G);
Regular extensions
Data not computed