LMFDB - Galois group: 26T34

Show commands: Magma

magma: G := TransitiveGroup(26, 34);

Group invariants

Abstract group:	$C_{13}^2:(C_3\times Q_8)$	magma: IdentifyGroup(G);
Order:	$4056=2^{3} \cdot 3 \cdot 13^{2}$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent	magma: NilpotencyClass(G);

Group action invariants

Degree $n$:	$26$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$34$	magma: t, n := TransitiveGroupIdentification(G); t;
Parity:	$-1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
$\card{\Aut(F/K)}$:	$1$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,26,8,17,10,20,5,19,11,15,9,25)(2,21,12,23,13,18,4,24,7,22,6,14)(3,16)$, $(1,6,9,3,2,4,13,8,5,11,12,10)(14,20,19,17,26,18,15,22,23,25,16,24)$	magma: Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 3
$3$: $C_3$
$4$: $C_2^2$
$6$: $C_6$ x 3
$8$: $Q_8$
$12$: $C_6\times C_2$
$24$: 24T4

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$3$:	$C_3$
$4$:	$C_2^2$
$6$:	$C_6$ x 3
$8$:	$Q_8$
$12$:	$C_6\times C_2$
$24$:	24T4

Subfields

Degree 2: $C_2$
Degree 13: None

Low degree siblings

26T34 x 2
Siblings 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	$2^{12},1^{2}$	$169$	$2$	$12$	$( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(14,15)(16,26)(17,25)(18,24)(19,23)(20,22)$
3A1	$3^{8},1^{2}$	$169$	$3$	$16$	$( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(14,26,23)(15,16,19)(17,22,24)(18,25,20)$
3A-1	$3^{8},1^{2}$	$169$	$3$	$16$	$( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(14,23,26)(15,19,16)(17,24,22)(18,20,25)$
4A	$4^{6},2$	$338$	$4$	$19$	$( 1,24, 7,22)( 2,15, 6,18)( 3,19, 5,14)( 4,23)( 8,26,13,20)( 9,17,12,16)(10,21,11,25)$
4B	$4^{6},2$	$338$	$4$	$19$	$( 1,24, 5,26)( 2,18, 4,19)( 3,25)( 6,20,13,17)( 7,14,12,23)( 8,21,11,16)( 9,15,10,22)$
4C	$4^{6},1^{2}$	$338$	$4$	$18$	$( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(14,25,15,17)(16,22,26,20)(18,19,24,23)$
6A1	$6^{4},1^{2}$	$169$	$6$	$20$	$( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(14,19,26,15,23,16)(17,18,22,25,24,20)$
6A-1	$6^{4},1^{2}$	$169$	$6$	$20$	$( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(14,16,23,15,26,19)(17,20,24,25,22,18)$
12A1	$12^{2},2$	$338$	$12$	$23$	$( 1,24,12,26, 4,21,11,14,13,25, 8,17)( 2,23, 3,22, 7,18,10,15, 9,16, 5,20)( 6,19)$
12A-1	$12^{2},2$	$338$	$12$	$23$	$( 1,24, 9,21,11,17, 5,16,10,19, 8,23)( 2,22, 6,14, 7,25, 4,18,13,26,12,15)( 3,20)$
12B1	$12^{2},1^{2}$	$338$	$12$	$22$	$( 2, 8,11, 6,10,12,13, 7, 4, 9, 5, 3)(14,18,16,17,23,20,15,24,26,25,19,22)$
12B-1	$12^{2},1^{2}$	$338$	$12$	$22$	$( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(14,20,19,17,26,18,15,22,23,25,16,24)$
12C1	$12^{2},2$	$338$	$12$	$23$	$( 1,24)( 2,21,11,20,10,23,13,14, 4,15, 5,25)( 3,18, 8,16, 6,22,12,17, 7,19, 9,26)$
12C-1	$12^{2},2$	$338$	$12$	$23$	$( 1,24, 6,25,13,16, 2,19,10,18, 3,14)( 4,22, 5,17, 9,23,12,21,11,26, 7,20)( 8,15)$
13A	$13^{2}$	$24$	$13$	$24$	$( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
13B	$13,1^{13}$	$24$	$13$	$12$	$( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)$
13C	$13^{2}$	$24$	$13$	$24$	$( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13D1	$13^{2}$	$24$	$13$	$24$	$( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,17,20,23,26,16,19,22,25,15,18,21,24)$
13D2	$13^{2}$	$24$	$13$	$24$	$( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,22,17,25,20,15,23,18,26,21,16,24,19)$
13E1	$13^{2}$	$24$	$13$	$24$	$( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,17,20,23,26,16,19,22,25,15,18,21,24)$
13E2	$13^{2}$	$24$	$13$	$24$	$( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,20,26,19,25,18,24,17,23,16,22,15,21)$