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Magma
magma: G := TransitiveGroup(26, 23);
Group invariants
| Abstract group: | $C_{13}^2:D_6$ | magma: IdentifyGroup(G);
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| Order: | $2028=2^{2} \cdot 3 \cdot 13^{2}$ | 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 $t$: | $23$ | 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: | $(1,11,12,3,6,5)(4,10,8,13,7,9)(14,15,25,21,20,23)(16,22,17,19,26,18)$, $(1,20,12,16,10,25,8,21,6,17,4,26,2,22,13,18,11,14,9,23,7,19,5,15,3,24)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $12$: $D_{6}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: None
Low degree siblings
39T34 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 | $2^{13}$ | $39$ | $2$ | $13$ | $( 1,20)( 2,22)( 3,24)( 4,26)( 5,15)( 6,17)( 7,19)( 8,21)( 9,23)(10,25)(11,14)(12,16)(13,18)$ |
| 2B | $2^{13}$ | $39$ | $2$ | $13$ | $( 1,26)( 2,20)( 3,14)( 4,21)( 5,15)( 6,22)( 7,16)( 8,23)( 9,17)(10,24)(11,18)(12,25)(13,19)$ |
| 2C | $2^{12},1^{2}$ | $169$ | $2$ | $12$ | $( 1, 2)( 3,13)( 4,12)( 5,11)( 6,10)( 7, 9)(14,15)(16,26)(17,25)(18,24)(19,23)(20,22)$ |
| 3A | $3^{8},1^{2}$ | $338$ | $3$ | $16$ | $( 1,10,13)( 2, 6, 3)( 4,11, 9)( 5, 7,12)(14,26,23)(15,16,19)(17,22,24)(18,25,20)$ |
| 6A | $6^{4},1^{2}$ | $338$ | $6$ | $20$ | $( 1, 3,10, 2,13, 6)( 4, 7,11,12, 9, 5)(14,19,26,15,23,16)(17,18,22,25,24,20)$ |
| 13A1 | $13^{2}$ | $6$ | $13$ | $24$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,16,18,20,22,24,26,15,17,19,21,23,25)$ |
| 13A2 | $13^{2}$ | $6$ | $13$ | $24$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,24,21,18,15,25,22,19,16,26,23,20,17)$ |
| 13A3 | $13^{2}$ | $6$ | $13$ | $24$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,26,25,24,23,22,21,20,19,18,17,16,15)$ |
| 13A4 | $13^{2}$ | $6$ | $13$ | $24$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,25,23,21,19,17,15,26,24,22,20,18,16)$ |
| 13A5 | $13^{2}$ | $6$ | $13$ | $24$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,17,20,23,26,16,19,22,25,15,18,21,24)$ |
| 13A6 | $13^{2}$ | $6$ | $13$ | $24$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,20,26,19,25,18,24,17,23,16,22,15,21)$ |
| 13B1 | $13^{2}$ | $6$ | $13$ | $24$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,21,15,22,16,23,17,24,18,25,19,26,20)$ |
| 13B2 | $13^{2}$ | $6$ | $13$ | $24$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,15,16,17,18,19,20,21,22,23,24,25,26)$ |
| 13B3 | $13^{2}$ | $6$ | $13$ | $24$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,20,26,19,25,18,24,17,23,16,22,15,21)$ |
| 13B4 | $13^{2}$ | $6$ | $13$ | $24$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,25,23,21,19,17,15,26,24,22,20,18,16)$ |
| 13B5 | $13^{2}$ | $6$ | $13$ | $24$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,18,22,26,17,21,25,16,20,24,15,19,23)$ |
| 13B6 | $13^{2}$ | $6$ | $13$ | $24$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,17,20,23,26,16,19,22,25,15,18,21,24)$ |
| 13C1 | $13,1^{13}$ | $12$ | $13$ | $12$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)$ |
| 13C2 | $13,1^{13}$ | $12$ | $13$ | $12$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)$ |
| 13D1 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,16,18,20,22,24,26,15,17,19,21,23,25)$ |
| 13D2 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,23,19,15,24,20,16,25,21,17,26,22,18)$ |
| 13D3 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,20,26,19,25,18,24,17,23,16,22,15,21)$ |
| 13D4 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,19,24,16,21,26,18,23,15,20,25,17,22)$ |
| 13D5 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,24,21,18,15,25,22,19,16,26,23,20,17)$ |
| 13D6 | $13^{2}$ | $12$ | $13$ | $24$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,15,16,17,18,19,20,21,22,23,24,25,26)$ |
| 26A1 | $26$ | $78$ | $26$ | $25$ | $( 1,21, 2,23, 3,25, 4,14, 5,16, 6,18, 7,20, 8,22, 9,24,10,26,11,15,12,17,13,19)$ |
| 26A3 | $26$ | $78$ | $26$ | $25$ | $( 1,23, 4,16, 7,22,10,15,13,21, 3,14, 6,20, 9,26,12,19, 2,25, 5,18, 8,24,11,17)$ |
| 26A5 | $26$ | $78$ | $26$ | $25$ | $( 1,25, 6,22,11,19, 3,16, 8,26,13,23, 5,20,10,17, 2,14, 7,24,12,21, 4,18, 9,15)$ |
| 26A7 | $26$ | $78$ | $26$ | $25$ | $( 1,14, 8,15, 2,16, 9,17, 3,18,10,19, 4,20,11,21, 5,22,12,23, 6,24,13,25, 7,26)$ |
| 26A9 | $26$ | $78$ | $26$ | $25$ | $( 1,16,10,21, 6,26, 2,18,11,23, 7,15, 3,20,12,25, 8,17, 4,22,13,14, 9,19, 5,24)$ |
| 26A11 | $26$ | $78$ | $26$ | $25$ | $( 1,18,12,14,10,23, 8,19, 6,15, 4,24, 2,20,13,16,11,25, 9,21, 7,17, 5,26, 3,22)$ |
| 26B1 | $26$ | $78$ | $26$ | $25$ | $( 1,25,10,23, 6,21, 2,19,11,17, 7,15, 3,26,12,24, 8,22, 4,20,13,18, 9,16, 5,14)$ |
| 26B3 | $26$ | $78$ | $26$ | $25$ | $( 1,23, 2,17, 3,24, 4,18, 5,25, 6,19, 7,26, 8,20, 9,14,10,21,11,15,12,22,13,16)$ |
| 26B5 | $26$ | $78$ | $26$ | $25$ | $( 1,21, 7,24,13,14, 6,17,12,20, 5,23,11,26, 4,16,10,19, 3,22, 9,25, 2,15, 8,18)$ |
| 26B7 | $26$ | $78$ | $26$ | $25$ | $( 1,19,12,18,10,17, 8,16, 6,15, 4,14, 2,26,13,25,11,24, 9,23, 7,22, 5,21, 3,20)$ |
| 26B9 | $26$ | $78$ | $26$ | $25$ | $( 1,17, 4,25, 7,20,10,15,13,23, 3,18, 6,26, 9,21,12,16, 2,24, 5,19, 8,14,11,22)$ |
| 26B11 | $26$ | $78$ | $26$ | $25$ | $( 1,15, 9,19, 4,23,12,14, 7,18, 2,22,10,26, 5,17,13,21, 8,25, 3,16,11,20, 6,24)$ |
Malle's constant $a(G)$: $1/12$
magma: ConjugacyClasses(G);
Character table
38 x 38 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed