LMFDB - Galois group: 44T140

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magma:G := TransitiveGroup(44, 140);

Group invariants

Abstract group:	$C_2\times M_{11}$	magma:IdentifyGroup(G);
Order:	$15840=2^{5} \cdot 3^{2} \cdot 5 \cdot 11$	magma:Order(G);
Cyclic:	no	magma:IsCyclic(G);
Abelian:	no	magma:IsAbelian(G);
Solvable:	no	magma:IsSolvable(G);
Nilpotency class:	not nilpotent	magma:NilpotencyClass(G);

Group action invariants

Degree $n$:	$44$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$140$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$4$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,18,6,9,27,13)(2,17,5,10,28,14)(3,20,7,11,25,16)(4,19,8,12,26,15)(21,37,44,22,38,43)(23,40,42,24,39,41)(29,35)(30,36)(31,33)(32,34)$, $(1,44,36,14,39,23,12,17,32,6,25)(2,43,35,13,40,24,11,18,31,5,26)(3,42,33,15,38,21,10,19,29,7,27)(4,41,34,16,37,22,9,20,30,8,28)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$
$7920$: $M_{11}$

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$7920$:	$M_{11}$

Subfields

Degree 2: $C_2$
Degree 4: None
Degree 11: $M_{11}$
Degree 22: $M_{11}$, 22T26, 22T27

Low degree siblings

22T26, 22T27, 24T12204
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	$1^{44}$	$1$	$1$	$0$	$()$
2A	$2^{22}$	$1$	$2$	$22$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)$
2B	$2^{16},1^{12}$	$165$	$2$	$16$	$( 1,10)( 2, 9)( 3,12)( 4,11)(13,37)(14,38)(15,39)(16,40)(21,33)(22,34)(23,36)(24,35)(25,42)(26,41)(27,44)(28,43)$
2C	$2^{22}$	$165$	$2$	$22$	$( 1, 2)( 3, 4)( 5,15)( 6,16)( 7,13)( 8,14)( 9,17)(10,18)(11,19)(12,20)(21,30)(22,29)(23,31)(24,32)(25,26)(27,28)(33,40)(34,39)(35,38)(36,37)(41,42)(43,44)$
3A	$3^{12},1^{8}$	$440$	$3$	$24$	$( 1,12,27)( 2,11,28)( 3,10,25)( 4, 9,26)( 5,16,20)( 6,15,19)( 7,14,17)( 8,13,18)(21,36,44)(22,35,43)(23,33,42)(24,34,41)$
4A	$4^{8},2^{4},1^{4}$	$990$	$4$	$28$	$( 1,23,10,36)( 2,24, 9,35)( 3,21,12,33)( 4,22,11,34)( 5, 8)( 6, 7)(13,43,37,28)(14,44,38,27)(15,42,39,25)(16,41,40,26)(17,19)(18,20)$
4B	$4^{8},2^{6}$	$990$	$4$	$30$	$( 1,22,17,31)( 2,21,18,32)( 3,24,19,30)( 4,23,20,29)( 5, 7)( 6, 8)( 9,15,43,36)(10,16,44,35)(11,14,41,33)(12,13,42,34)(25,28)(26,27)(37,38)(39,40)$
5A	$5^{8},1^{4}$	$1584$	$5$	$32$	$( 1, 6,27,21,38)( 2, 5,28,22,37)( 3, 7,25,23,39)( 4, 8,26,24,40)( 9,31,13,34,18)(10,32,14,33,17)(11,30,16,35,20)(12,29,15,36,19)$
6A	$6^{6},2^{4}$	$440$	$6$	$34$	$( 1,30, 6, 2,29, 5)( 3,31, 7, 4,32, 8)( 9,25,41,10,26,42)(11,27,43,12,28,44)(13,14)(15,16)(17,34,39,18,33,40)(19,35,38,20,36,37)(21,22)(23,24)$
6B	$6^{4},3^{4},2^{4}$	$1320$	$6$	$32$	$( 1, 7,12,14,27,17)( 2, 8,11,13,28,18)( 3, 6,10,15,25,19)( 4, 5, 9,16,26,20)(21,44,36)(22,43,35)(23,42,33)(24,41,34)(29,38)(30,37)(31,40)(32,39)$
6C	$6^{6},2^{4}$	$1320$	$6$	$34$	$( 1,28,44, 2,27,43)( 3,26,42, 4,25,41)( 5,36,22,15,37,29)( 6,35,21,16,38,30)( 7,34,23,13,39,31)( 8,33,24,14,40,32)( 9,17)(10,18)(11,19)(12,20)$
8A1	$8^{4},4^{2},2^{2}$	$990$	$8$	$36$	$( 1,37,23,28,10,13,36,43)( 2,38,24,27, 9,14,35,44)( 3,40,21,26,12,16,33,41)( 4,39,22,25,11,15,34,42)( 5,19, 8,17)( 6,20, 7,18)(29,31)(30,32)$
8A-1	$8^{4},4^{2},2^{2}$	$990$	$8$	$36$	$( 1,43,36,13,10,28,23,37)( 2,44,35,14, 9,27,24,38)( 3,41,33,16,12,26,21,40)( 4,42,34,15,11,25,22,39)( 5,17, 8,19)( 6,18, 7,20)(29,31)(30,32)$
8B1	$8^{4},4^{2},2^{2}$	$990$	$8$	$36$	$( 1,10,15,32,17,44,36,21)( 2, 9,16,31,18,43,35,22)( 3,12,14,29,19,42,33,23)( 4,11,13,30,20,41,34,24)( 5, 8)( 6, 7)(25,38,27,39)(26,37,28,40)$
8B-1	$8^{4},4^{2},2^{2}$	$990$	$8$	$36$	$( 1,21,36,44,17,32,15,10)( 2,22,35,43,18,31,16, 9)( 3,23,33,42,19,29,14,12)( 4,24,34,41,20,30,13,11)( 5, 8)( 6, 7)(25,39,27,38)(26,40,28,37)$
10A	$10^{4},2^{2}$	$1584$	$10$	$38$	$( 1,22, 6,37,27, 2,21, 5,38,28)( 3,24, 7,40,25, 4,23, 8,39,26)( 9,33,31,17,13,10,34,32,18,14)(11,36,30,19,16,12,35,29,20,15)(41,42)(43,44)$
11A1	$11^{4}$	$720$	$11$	$40$	$( 1,25,44,17,32,36,39,23,14,12, 6)( 2,26,43,18,31,35,40,24,13,11, 5)( 3,27,42,19,29,33,38,21,15,10, 7)( 4,28,41,20,30,34,37,22,16, 9, 8)$
11A-1	$11^{4}$	$720$	$11$	$40$	$( 1, 6,12,14,23,39,36,32,17,44,25)( 2, 5,11,13,24,40,35,31,18,43,26)( 3, 7,10,15,21,38,33,29,19,42,27)( 4, 8, 9,16,22,37,34,30,20,41,28)$
22A1	$22^{2}$	$720$	$22$	$42$	$( 1,40,25,24,44,13,17,11,32, 5,36, 2,39,26,23,43,14,18,12,31, 6,35)( 3,37,27,22,42,16,19, 9,29, 8,33, 4,38,28,21,41,15,20,10,30, 7,34)$
22A-1	$22^{2}$	$720$	$22$	$42$	$( 1,35, 6,31,12,18,14,43,23,26,39, 2,36, 5,32,11,17,13,44,24,25,40)( 3,34, 7,30,10,20,15,41,21,28,38, 4,33, 8,29, 9,19,16,42,22,27,37)$

Malle's constant $a(G)$: $1/16$

magma:ConjugacyClasses(G);

Character table

		1A	2A	2B	2C	3A	4A	4B	5A	6A	6B	6C	8A1	8A-1	8B1	8B-1	10A	11A1	11A-1	22A1	22A-1
	Size	1	1	165	165	440	990	990	1584	440	1320	1320	990	990	990	990	1584	720	720	720	720
	2 P	1A	1A	1A	1A	3A	2B	2B	5A	3A	3A	3A	4A	4A	4A	4A	5A	11A-1	11A1	11A1	11A-1
	3 P	1A	2A	2B	2C	1A	4A	4B	5A	2A	2B	2C	8A1	8A-1	8B1	8B-1	10A	11A1	11A-1	22A1	22A-1
	5 P	1A	2A	2B	2C	3A	4A	4B	1A	6A	6B	6C	8A-1	8A1	8B-1	8B1	2A	11A1	11A-1	22A1	22A-1
	11 P	1A	2A	2B	2C	3A	4A	4B	5A	6A	6B	6C	8A1	8A-1	8B1	8B-1	10A	1A	1A	2A	2A
	Type
15840.q.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
15840.q.1b	R	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$
15840.q.10a	R	$10$	$10$	$2$	$2$	$1$	$2$	$2$	$0$	$1$	$- 1$	$- 1$	$0$	$0$	$0$	$0$	$0$	$- 1$	$- 1$	$- 1$	$- 1$
15840.q.10b	R	$10$	$- 10$	$2$	$- 2$	$1$	$2$	$- 2$	$0$	$- 1$	$- 1$	$1$	$0$	$0$	$0$	$0$	$0$	$- 1$	$- 1$	$1$	$1$
15840.q.10c1	C	$10$	$- 10$	$- 2$	$2$	$1$	$0$	$0$	$0$	$- 1$	$1$	$- 1$	$- ζ_{8} - ζ_{8}^{3}$	$ζ_{8} + ζ_{8}^{3}$	$- ζ_{8} - ζ_{8}^{3}$	$ζ_{8} + ζ_{8}^{3}$	$0$	$- 1$	$- 1$	$1$	$1$
15840.q.10c2	C	$10$	$- 10$	$- 2$	$2$	$1$	$0$	$0$	$0$	$- 1$	$1$	$- 1$	$ζ_{8} + ζ_{8}^{3}$	$- ζ_{8} - ζ_{8}^{3}$	$ζ_{8} + ζ_{8}^{3}$	$- ζ_{8} - ζ_{8}^{3}$	$0$	$- 1$	$- 1$	$1$	$1$
15840.q.10d1	C	$10$	$10$	$- 2$	$- 2$	$1$	$0$	$0$	$0$	$1$	$1$	$1$	$- ζ_{8} - ζ_{8}^{3}$	$- ζ_{8} - ζ_{8}^{3}$	$ζ_{8} + ζ_{8}^{3}$	$ζ_{8} + ζ_{8}^{3}$	$0$	$- 1$	$- 1$	$- 1$	$- 1$
15840.q.10d2	C	$10$	$10$	$- 2$	$- 2$	$1$	$0$	$0$	$0$	$1$	$1$	$1$	$ζ_{8} + ζ_{8}^{3}$	$ζ_{8} + ζ_{8}^{3}$	$- ζ_{8} - ζ_{8}^{3}$	$- ζ_{8} - ζ_{8}^{3}$	$0$	$- 1$	$- 1$	$- 1$	$- 1$
15840.q.11a	R	$11$	$11$	$3$	$3$	$2$	$- 1$	$- 1$	$1$	$2$	$0$	$0$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$0$	$0$	$0$	$0$
15840.q.11b	R	$11$	$- 11$	$3$	$- 3$	$2$	$- 1$	$1$	$1$	$- 2$	$0$	$0$	$- 1$	$1$	$1$	$- 1$	$- 1$	$0$	$0$	$0$	$0$
15840.q.16a1	C	$16$	$16$	$0$	$0$	$- 2$	$0$	$0$	$1$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	$- ζ_{11}^{- 2} - 1 - ζ_{11} - ζ_{11}^{3} - ζ_{11}^{4} - ζ_{11}^{5}$	$ζ_{11}^{- 2} + ζ_{11} + ζ_{11}^{3} + ζ_{11}^{4} + ζ_{11}^{5}$	$ζ_{11}^{- 2} + ζ_{11} + ζ_{11}^{3} + ζ_{11}^{4} + ζ_{11}^{5}$	$- ζ_{11}^{- 2} - 1 - ζ_{11} - ζ_{11}^{3} - ζ_{11}^{4} - ζ_{11}^{5}$
15840.q.16a2	C	$16$	$16$	$0$	$0$	$- 2$	$0$	$0$	$1$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	$ζ_{11}^{- 2} + ζ_{11} + ζ_{11}^{3} + ζ_{11}^{4} + ζ_{11}^{5}$	$- ζ_{11}^{- 2} - 1 - ζ_{11} - ζ_{11}^{3} - ζ_{11}^{4} - ζ_{11}^{5}$	$- ζ_{11}^{- 2} - 1 - ζ_{11} - ζ_{11}^{3} - ζ_{11}^{4} - ζ_{11}^{5}$	$ζ_{11}^{- 2} + ζ_{11} + ζ_{11}^{3} + ζ_{11}^{4} + ζ_{11}^{5}$
15840.q.16b1	C	$16$	$- 16$	$0$	$0$	$- 2$	$0$	$0$	$1$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$	$- ζ_{11}^{- 2} - 1 - ζ_{11} - ζ_{11}^{3} - ζ_{11}^{4} - ζ_{11}^{5}$	$ζ_{11}^{- 2} + ζ_{11} + ζ_{11}^{3} + ζ_{11}^{4} + ζ_{11}^{5}$	$- ζ_{11}^{- 2} - ζ_{11} - ζ_{11}^{3} - ζ_{11}^{4} - ζ_{11}^{5}$	$ζ_{11}^{- 2} + 1 + ζ_{11} + ζ_{11}^{3} + ζ_{11}^{4} + ζ_{11}^{5}$
15840.q.16b2	C	$16$	$- 16$	$0$	$0$	$- 2$	$0$	$0$	$1$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$	$ζ_{11}^{- 2} + ζ_{11} + ζ_{11}^{3} + ζ_{11}^{4} + ζ_{11}^{5}$	$- ζ_{11}^{- 2} - 1 - ζ_{11} - ζ_{11}^{3} - ζ_{11}^{4} - ζ_{11}^{5}$	$ζ_{11}^{- 2} + 1 + ζ_{11} + ζ_{11}^{3} + ζ_{11}^{4} + ζ_{11}^{5}$	$- ζ_{11}^{- 2} - ζ_{11} - ζ_{11}^{3} - ζ_{11}^{4} - ζ_{11}^{5}$
15840.q.44a	R	$44$	$44$	$4$	$4$	$- 1$	$0$	$0$	$- 1$	$- 1$	$1$	$1$	$0$	$0$	$0$	$0$	$- 1$	$0$	$0$	$0$	$0$
15840.q.44b	R	$44$	$- 44$	$4$	$- 4$	$- 1$	$0$	$0$	$- 1$	$1$	$1$	$- 1$	$0$	$0$	$0$	$0$	$1$	$0$	$0$	$0$	$0$
15840.q.45a	R	$45$	$- 45$	$- 3$	$3$	$0$	$1$	$- 1$	$0$	$0$	$0$	$0$	$- 1$	$1$	$1$	$- 1$	$0$	$1$	$1$	$- 1$	$- 1$
15840.q.45b	R	$45$	$45$	$- 3$	$- 3$	$0$	$1$	$1$	$0$	$0$	$0$	$0$	$- 1$	$- 1$	$- 1$	$- 1$	$0$	$1$	$1$	$1$	$1$
15840.q.55a	R	$55$	$- 55$	$- 1$	$1$	$1$	$- 1$	$1$	$0$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$0$	$0$	$0$	$0$	$0$
15840.q.55b	R	$55$	$55$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$0$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$0$	$0$	$0$	$0$	$0$

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	44T140

Degree	$44$
Order	$15840$
Cyclic	no
Abelian	no
Solvable	no
Primitive	no
$p$-group	no
Group:	$C_2\times M_{11}$