LMFDB - Galois group: 26T44

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magma: G := TransitiveGroup(26, 44);

Group invariants

Abstract group:	$C_{13}^2:C_3:\OD_{16}$	magma: IdentifyGroup(G);
Order:	$8112=2^{4} \cdot 3 \cdot 13^{2}$	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 $t$:	$44$	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:	$(1,9,11,5,10,8)(2,6,7,4,13,12)(14,15,24)(16,20,17)(18,25,23)(19,21,26)$, $(1,17,3,24,13,20,11,26)(2,14,8,22,12,23,6,15)(4,21,5,18,10,16,9,19)(7,25)$	magma: Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_4$ x 2, $C_2^2$
$6$: $S_3$
$8$: $C_4\times C_2$
$12$: $D_{6}$, $C_3 : C_4$ x 2
$16$: $C_8:C_2$
$24$: 24T6
$48$: 24T20

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$6$:	$S_3$
$8$:	$C_4\times C_2$
$12$:	$D_{6}$, $C_3 : C_4$ x 2
$16$:	$C_8:C_2$
$24$:	24T6
$48$:	24T20

Subfields

Degree 2: $C_2$
Degree 13: None

Low degree siblings

There are no siblings 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	$2^{6},1^{14}$	$26$	$2$	$6$	$(14,23)(15,22)(16,21)(17,20)(18,19)(24,26)$
2B	$2^{12},1^{2}$	$169$	$2$	$12$	$( 1, 5)( 2, 4)( 6,13)( 7,12)( 8,11)( 9,10)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$
3A	$3^{8},1^{2}$	$338$	$3$	$16$	$( 1,13, 4)( 2, 9, 7)( 3, 5,10)( 8,11,12)(14,18,17)(15,21,26)(16,24,22)(19,20,23)$
4A1	$4^{6},1^{2}$	$169$	$4$	$18$	$( 1, 6, 5,13)( 2,11, 4, 8)( 7,10,12, 9)(15,19,26,22)(16,24,25,17)(18,21,23,20)$
4A-1	$4^{6},1^{2}$	$169$	$4$	$18$	$( 1,13, 5, 6)( 2, 8, 4,11)( 7, 9,12,10)(15,22,26,19)(16,17,25,24)(18,20,23,21)$
4B	$4^{6},1^{2}$	$338$	$4$	$18$	$( 1,12, 9,11)( 2, 7, 8, 3)( 4,10, 6,13)(14,23,16,20)(17,25,26,18)(19,22,24,21)$
6A	$6^{4},1^{2}$	$338$	$6$	$20$	$( 1, 2, 6, 9, 8, 4)( 3,10,12, 7,13,11)(14,18,19,16,25,24)(17,22,20,26,21,23)$
6B1	$6^{2},3^{4},1^{2}$	$338$	$6$	$18$	$( 1, 4,13)( 2, 7, 9)( 3,10, 5)( 8,12,11)(14,20,18,23,17,19)(15,24,21,22,26,16)$
6B-1	$6^{2},3^{4},1^{2}$	$338$	$6$	$18$	$( 1,13, 4)( 2, 9, 7)( 3, 5,10)( 8,11,12)(14,19,17,23,18,20)(15,16,26,22,21,24)$
8A1	$8^{3},2$	$1014$	$8$	$22$	$( 1,24, 6,25, 5,17,13,16)( 2,19,11,26, 4,22, 8,15)( 3,14)( 7,20,10,18,12,21, 9,23)$
8A-1	$8^{3},2$	$1014$	$8$	$22$	$( 1,16,13,17, 5,25, 6,24)( 2,15, 8,22, 4,26,11,19)( 3,14)( 7,23, 9,21,12,18,10,20)$
8B1	$8^{3},2$	$1014$	$8$	$22$	$( 1,15,12,23, 2,24, 4,16)( 3,20, 9,22,13,19, 7,17)( 5,25, 6,21,11,14,10,18)( 8,26)$
8B-1	$8^{3},2$	$1014$	$8$	$22$	$( 1,16, 4,24, 2,23,12,15)( 3,17, 7,19,13,22, 9,20)( 5,18,10,14,11,21, 6,25)( 8,26)$
12A1	$12^{2},1^{2}$	$338$	$12$	$22$	$( 1,10, 2,12, 6, 7, 9,13, 8,11, 4, 3)(14,21,18,23,19,17,16,22,25,20,24,26)$
12A5	$12^{2},1^{2}$	$338$	$12$	$22$	$( 1, 7, 4,12, 8,10, 9, 3, 6,11, 2,13)(14,17,24,23,25,21,16,26,19,20,18,22)$
12B1	$12^{2},1^{2}$	$338$	$12$	$22$	$( 1, 9,12, 5, 4, 2,11, 3,13, 7, 8,10)(14,17,22,26,24,25,18,15,23,19,21,20)$
12B-1	$12^{2},1^{2}$	$338$	$12$	$22$	$( 1,10, 8, 7,13, 3,11, 2, 4, 5,12, 9)(14,20,21,19,23,15,18,25,24,26,22,17)$
13A	$13,1^{13}$	$24$	$13$	$12$	$(14,17,20,23,26,16,19,22,25,15,18,21,24)$
13B1	$13^{2}$	$48$	$13$	$24$	$( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,17,20,23,26,16,19,22,25,15,18,21,24)$
13B2	$13^{2}$	$48$	$13$	$24$	$( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,20,26,19,25,18,24,17,23,16,22,15,21)$
13B4	$13^{2}$	$48$	$13$	$24$	$( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,20,26,19,25,18,24,17,23,16,22,15,21)$
26A	$13,2^{6},1$	$312$	$26$	$18$	$( 1, 5)( 2, 4)( 6,13)( 7,12)( 8,11)( 9,10)(14,22,17,25,20,15,23,18,26,21,16,24,19)$

Malle's constant $a(G)$: $1/6$

magma: ConjugacyClasses(G);

Character table

		1A	2A	2B	3A	4A1	4A-1	4B	6A	6B1	6B-1	8A1	8A-1	8B1	8B-1	12A1	12A5	12B1	12B-1	13A	13B1	13B2	13B4	26A
	Size	1	26	169	338	169	169	338	338	338	338	1014	1014	1014	1014	338	338	338	338	24	48	48	48	312
	2 P	1A	1A	1A	3A	2B	2B	2B	3A	3A	3A	4A1	4A-1	4A1	4A-1	6A	6A	6A	6A	13A	13B2	13B4	13B1	13A
	3 P	1A	2A	2B	1A	4A-1	4A1	4B	2B	2A	2A	8A-1	8A1	8B-1	8B1	4B	4B	4A-1	4A1	13A	13B2	13B4	13B1	26A
	13 P	1A	2A	2B	3A	4A1	4A-1	4B	6A	6B1	6B-1	8A1	8A-1	8B1	8B-1	12A1	12A5	12B1	12B-1	1A	1A	1A	1A	2A
	Type
8112.be.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
8112.be.1b	R	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$
8112.be.1c	R	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$
8112.be.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
8112.be.1e1	C	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- i$	$i$	$- i$	$i$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$- 1$
8112.be.1e2	C	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$i$	$- i$	$i$	$- i$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$- 1$
8112.be.1f1	C	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$- i$	$i$	$i$	$- i$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$
8112.be.1f2	C	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$i$	$- i$	$- i$	$i$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$
8112.be.2a	R	$2$	$2$	$2$	$- 1$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$0$	$0$	$0$	$0$	$- 1$	$- 1$	$- 1$	$- 1$	$2$	$2$	$2$	$2$	$2$
8112.be.2b	R	$2$	$- 2$	$2$	$- 1$	$2$	$2$	$- 2$	$- 1$	$1$	$1$	$0$	$0$	$0$	$0$	$- 1$	$1$	$- 1$	$1$	$2$	$2$	$2$	$2$	$- 2$
8112.be.2c	S	$2$	$- 2$	$2$	$- 1$	$- 2$	$- 2$	$2$	$- 1$	$1$	$1$	$0$	$0$	$0$	$0$	$1$	$- 1$	$1$	$- 1$	$2$	$2$	$2$	$2$	$- 2$
8112.be.2d	S	$2$	$2$	$2$	$- 1$	$- 2$	$- 2$	$- 2$	$- 1$	$- 1$	$- 1$	$0$	$0$	$0$	$0$	$1$	$1$	$1$	$1$	$2$	$2$	$2$	$2$	$2$
8112.be.2e1	C	$2$	$0$	$- 2$	$2$	$- 2 i$	$2 i$	$0$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2 i$	$0$	$2 i$	$0$	$2$	$2$	$2$	$2$	$0$
8112.be.2e2	C	$2$	$0$	$- 2$	$2$	$2 i$	$- 2 i$	$0$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$2 i$	$0$	$- 2 i$	$0$	$2$	$2$	$2$	$2$	$0$
8112.be.2f1	C	$2$	$0$	$- 2$	$- 1$	$- 2 ζ_{12}^{3}$	$2 ζ_{12}^{3}$	$0$	$1$	$1 - 2 ζ_{12}^{2}$	$- 1 + 2 ζ_{12}^{2}$	$0$	$0$	$0$	$0$	$ζ_{12}^{3}$	$ζ_{12}^{- 1} + ζ_{12}$	$- ζ_{12}^{3}$	$- ζ_{12}^{- 1} - ζ_{12}$	$2$	$2$	$2$	$2$	$0$
8112.be.2f2	C	$2$	$0$	$- 2$	$- 1$	$2 ζ_{12}^{3}$	$- 2 ζ_{12}^{3}$	$0$	$1$	$- 1 + 2 ζ_{12}^{2}$	$1 - 2 ζ_{12}^{2}$	$0$	$0$	$0$	$0$	$- ζ_{12}^{3}$	$ζ_{12}^{- 1} + ζ_{12}$	$ζ_{12}^{3}$	$- ζ_{12}^{- 1} - ζ_{12}$	$2$	$2$	$2$	$2$	$0$
8112.be.2f3	C	$2$	$0$	$- 2$	$- 1$	$- 2 ζ_{12}^{3}$	$2 ζ_{12}^{3}$	$0$	$1$	$- 1 + 2 ζ_{12}^{2}$	$1 - 2 ζ_{12}^{2}$	$0$	$0$	$0$	$0$	$ζ_{12}^{3}$	$- ζ_{12}^{- 1} - ζ_{12}$	$- ζ_{12}^{3}$	$ζ_{12}^{- 1} + ζ_{12}$	$2$	$2$	$2$	$2$	$0$
8112.be.2f4	C	$2$	$0$	$- 2$	$- 1$	$2 ζ_{12}^{3}$	$- 2 ζ_{12}^{3}$	$0$	$1$	$1 - 2 ζ_{12}^{2}$	$- 1 + 2 ζ_{12}^{2}$	$0$	$0$	$0$	$0$	$- ζ_{12}^{3}$	$- ζ_{12}^{- 1} - ζ_{12}$	$ζ_{12}^{3}$	$ζ_{12}^{- 1} + ζ_{12}$	$2$	$2$	$2$	$2$	$0$
8112.be.24a	R	$24$	$12$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$11$	$- 2$	$- 2$	$- 2$	$- 1$
8112.be.24b	R	$24$	$- 12$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$11$	$- 2$	$- 2$	$- 2$	$1$
8112.be.48a1	R	$48$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 4$	$4 ζ_{13}^{- 6} + 4 ζ_{13}^{- 4} + ζ_{13}^{- 3} + ζ_{13}^{- 2} + 2 + ζ_{13}^{2} + ζ_{13}^{3} + 4 ζ_{13}^{4} + 4 ζ_{13}^{6}$	$- 3 ζ_{13}^{- 6} - 3 ζ_{13}^{- 4} - 4 ζ_{13}^{- 3} - 4 ζ_{13}^{- 2} - 2 - 4 ζ_{13}^{2} - 4 ζ_{13}^{3} - 3 ζ_{13}^{4} - 3 ζ_{13}^{6}$	$- ζ_{13}^{- 6} - ζ_{13}^{- 4} + 3 ζ_{13}^{- 3} + 3 ζ_{13}^{- 2} + 1 + 3 ζ_{13}^{2} + 3 ζ_{13}^{3} - ζ_{13}^{4} - ζ_{13}^{6}$	$0$
8112.be.48a2	R	$48$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 4$	$- 3 ζ_{13}^{- 6} - 3 ζ_{13}^{- 4} - 4 ζ_{13}^{- 3} - 4 ζ_{13}^{- 2} - 2 - 4 ζ_{13}^{2} - 4 ζ_{13}^{3} - 3 ζ_{13}^{4} - 3 ζ_{13}^{6}$	$- ζ_{13}^{- 6} - ζ_{13}^{- 4} + 3 ζ_{13}^{- 3} + 3 ζ_{13}^{- 2} + 1 + 3 ζ_{13}^{2} + 3 ζ_{13}^{3} - ζ_{13}^{4} - ζ_{13}^{6}$	$4 ζ_{13}^{- 6} + 4 ζ_{13}^{- 4} + ζ_{13}^{- 3} + ζ_{13}^{- 2} + 2 + ζ_{13}^{2} + ζ_{13}^{3} + 4 ζ_{13}^{4} + 4 ζ_{13}^{6}$	$0$
8112.be.48a3	R	$48$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 4$	$- ζ_{13}^{- 6} - ζ_{13}^{- 4} + 3 ζ_{13}^{- 3} + 3 ζ_{13}^{- 2} + 1 + 3 ζ_{13}^{2} + 3 ζ_{13}^{3} - ζ_{13}^{4} - ζ_{13}^{6}$	$4 ζ_{13}^{- 6} + 4 ζ_{13}^{- 4} + ζ_{13}^{- 3} + ζ_{13}^{- 2} + 2 + ζ_{13}^{2} + ζ_{13}^{3} + 4 ζ_{13}^{4} + 4 ζ_{13}^{6}$	$- 3 ζ_{13}^{- 6} - 3 ζ_{13}^{- 4} - 4 ζ_{13}^{- 3} - 4 ζ_{13}^{- 2} - 2 - 4 ζ_{13}^{2} - 4 ζ_{13}^{3} - 3 ζ_{13}^{4} - 3 ζ_{13}^{6}$	$0$

magma: CharacterTable(G);

Regular extensions

Data not computed

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	26T44
Degree	$26$
Order	$8112$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_{13}^2:C_3:\OD_{16}$