LMFDB - Galois group: 44T46

Show commands: Magma

magma: G := TransitiveGroup(44, 46);

Group action invariants

Degree $n$:	$44$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$46$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$A_4\times F_{11}$
Parity:	$1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$1$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(1,20,13,28,36,11,37,43,29,22,2,17,15,27,33,9,38,42,31,21,3,19,16,26,35,10,39,44,32,24)(4,18,14,25,34,12,40,41,30,23)(6,8,7), (1,18,27,31,12,21,8,41,38,15,4,20,28,30,10,22,5,43,37,14,2,19,25,32,9,23,7,44,40,16)(3,17,26,29,11,24,6,42,39,13)(34,36,35)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$
$3$: $C_3$
$5$: $C_5$
$6$: $C_6$
$10$: $C_{10}$
$12$: $A_4$
$15$: $C_{15}$
$24$: $A_4\times C_2$
$30$: $C_{30}$
$60$: 20T14
$110$: $F_{11}$
$120$: 30T18
$330$: 33T8

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$5$:	$C_5$
$6$:	$C_6$
$10$:	$C_{10}$
$12$:	$A_4$
$15$:	$C_{15}$
$24$:	$A_4\times C_2$
$30$:	$C_{30}$
$60$:	20T14
$110$:	$F_{11}$
$120$:	30T18
$330$:	33T8

Subfields

Degree 2: None
Degree 4: $A_4$
Degree 11: $F_{11}$
Degree 22: 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	Representative
	$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 11, 11, 11, 11 $	$10$	$11$	$( 1,35,22, 9,44,31,19, 8,37,28,15)( 2,36,21,10,43,32,20, 7,38,27,16) ( 3,33,24,11,42,29,17, 6,39,26,13)( 4,34,23,12,41,30,18, 5,40,25,14)$
	$ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $	$11$	$5$	$( 5,14,40,23,18)( 6,13,39,24,17)( 7,16,38,21,20)( 8,15,37,22,19) ( 9,28,31,44,35)(10,27,32,43,36)(11,26,29,42,33)(12,25,30,41,34)$
	$ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $	$11$	$5$	$( 5,40,18,14,23)( 6,39,17,13,24)( 7,38,20,16,21)( 8,37,19,15,22) ( 9,31,35,28,44)(10,32,36,27,43)(11,29,33,26,42)(12,30,34,25,41)$
	$ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $	$11$	$5$	$( 5,18,23,40,14)( 6,17,24,39,13)( 7,20,21,38,16)( 8,19,22,37,15) ( 9,35,44,31,28)(10,36,43,32,27)(11,33,42,29,26)(12,34,41,30,25)$
	$ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $	$11$	$5$	$( 5,23,14,18,40)( 6,24,13,17,39)( 7,21,16,20,38)( 8,22,15,19,37) ( 9,44,28,35,31)(10,43,27,36,32)(11,42,26,33,29)(12,41,25,34,30)$
	$ 22, 22 $	$30$	$22$	$( 1,30,15,41,28,12,37,23, 8,34,19, 4,31,14,44,25, 9,40,22, 5,35,18) ( 2,29,16,42,27,11,38,24, 7,33,20, 3,32,13,43,26,10,39,21, 6,36,17)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$3$	$2$	$( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39)(41,44)(42,43)$
	$ 10, 10, 10, 10, 2, 2 $	$33$	$10$	$( 1,41,31,40,19, 4,44,30,37,18)( 2,42,32,39,20, 3,43,29,38,17)( 5, 9,23,15,34, 8,12,22,14,35)( 6,10,24,16,33, 7,11,21,13,36)(25,28)(26,27)$
	$ 10, 10, 10, 10, 2, 2 $	$33$	$10$	$( 1,34,15,12,19, 4,35,14, 9,18)( 2,33,16,11,20, 3,36,13,10,17)( 5,28,30,22,40, 8,25,31,23,37)( 6,27,29,21,39, 7,26,32,24,38)(41,44)(42,43)$
	$ 10, 10, 10, 10, 2, 2 $	$33$	$10$	$( 1,25,35,23,19, 4,28,34,22,18)( 2,26,36,24,20, 3,27,33,21,17)( 5,44,12,15,30, 8,41, 9,14,31)( 6,43,11,16,29, 7,42,10,13,32)(37,40)(38,39)$
	$ 10, 10, 10, 10, 2, 2 $	$33$	$10$	$( 1,12, 8,30,19, 4, 9, 5,31,18)( 2,11, 7,29,20, 3,10, 6,32,17)(13,27,42,36,39, 16,26,43,33,38)(14,28,41,35,40,15,25,44,34,37)(21,24)(22,23)$
	$ 15, 15, 5, 5, 3, 1 $	$44$	$15$	$( 2, 3, 4)( 5,38,17,14,21, 6,40,20,13,23, 7,39,18,16,24)( 8,37,19,15,22) ( 9,31,35,28,44)(10,29,34,27,42,12,32,33,25,43,11,30,36,26,41)$
	$ 15, 15, 5, 5, 3, 1 $	$44$	$15$	$( 2, 3, 4)( 5,21,13,18,38, 6,23,16,17,40, 7,24,14,20,39)( 8,22,15,19,37) ( 9,44,28,35,31)(10,42,25,36,29,12,43,26,34,32,11,41,27,33,30)$
	$ 15, 15, 5, 5, 3, 1 $	$44$	$15$	$( 2, 3, 4)( 5,20,24,40,16, 6,18,21,39,14, 7,17,23,38,13)( 8,19,22,37,15) ( 9,35,44,31,28)(10,33,41,32,26,12,36,42,30,27,11,34,43,29,25)$
	$ 15, 15, 5, 5, 3, 1 $	$44$	$15$	$( 2, 3, 4)( 5,16,39,23,20, 6,14,38,24,18, 7,13,40,21,17)( 8,15,37,22,19) ( 9,28,31,44,35)(10,26,30,43,33,12,27,29,41,36,11,25,32,42,34)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$4$	$3$	$( 2, 3, 4)( 5, 7, 6)(10,11,12)(13,14,16)(17,18,20)(21,24,23)(25,27,26) (29,30,32)(33,34,36)(38,39,40)(41,43,42)$
	$ 33, 11 $	$40$	$33$	$( 1,35,22, 9,44,31,19, 8,37,28,15)( 2,33,23,10,42,30,20, 6,40,27,13, 4,36,24, 12,43,29,18, 7,39,25,16, 3,34,21,11,41,32,17, 5,38,26,14)$
	$ 15, 15, 5, 5, 3, 1 $	$44$	$15$	$( 2, 4, 3)( 5,17,21,40,13, 7,18,24,38,14, 6,20,23,39,16)( 8,19,22,37,15) ( 9,35,44,31,28)(10,34,42,32,25,11,36,41,29,27,12,33,43,30,26)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$4$	$3$	$( 2, 4, 3)( 5, 6, 7)(10,12,11)(13,16,14)(17,20,18)(21,23,24)(25,26,27) (29,32,30)(33,36,34)(38,40,39)(41,42,43)$
	$ 33, 11 $	$40$	$33$	$( 1,35,22, 9,44,31,19, 8,37,28,15)( 2,34,24,10,41,29,20, 5,39,27,14, 3,36,23, 11,43,30,17, 7,40,26,16, 4,33,21,12,42,32,18, 6,38,25,13)$
	$ 15, 15, 5, 5, 3, 1 $	$44$	$15$	$( 2, 4, 3)( 5,13,38,23,17, 7,14,39,21,18, 6,16,40,24,20)( 8,15,37,22,19) ( 9,28,31,44,35)(10,25,29,43,34,11,27,30,42,36,12,26,32,41,33)$
	$ 15, 15, 5, 5, 3, 1 $	$44$	$15$	$( 2, 4, 3)( 5,24,16,18,39, 7,23,13,20,40, 6,21,14,17,38)( 8,22,15,19,37) ( 9,44,28,35,31)(10,41,26,36,30,11,43,25,33,32,12,42,27,34,29)$
	$ 15, 15, 5, 5, 3, 1 $	$44$	$15$	$( 2, 4, 3)( 5,39,20,14,24, 7,40,17,16,23, 6,38,18,13,21)( 8,37,19,15,22) ( 9,31,35,28,44)(10,30,33,27,41,11,32,34,26,43,12,29,36,25,42)$
	$ 30, 10, 3, 1 $	$44$	$30$	$( 2, 4, 3)( 5,33,38,25,17,43,14,11,21,30, 6,36,40,26,20,41,13,10,23,29, 7,34, 39,27,18,42,16,12,24,32)( 8,35,37,28,19,44,15, 9,22,31)$
	$ 30, 10, 3, 1 $	$44$	$30$	$( 2, 4, 3)( 5,11,20,34,24,43,40,29,16,25, 6,10,18,33,21,41,39,32,14,26, 7,12, 17,36,23,42,38,30,13,27)( 8, 9,19,35,22,44,37,31,15,28)$
	$ 30, 10, 3, 1 $	$44$	$30$	$( 2, 4, 3)( 5,26,16,30,39,43,23,33,20,12, 6,27,14,29,38,41,24,36,18,11, 7,25, 13,32,40,42,21,34,17,10)( 8,28,15,31,37,44,22,35,19, 9)$
	$ 6, 6, 6, 6, 6, 3, 2, 2, 2, 2, 2, 1 $	$44$	$6$	$( 2, 4, 3)( 5,42, 7,41, 6,43)( 8,44)( 9,37)(10,40,11,38,12,39)(13,36,14,33,16, 34)(15,35)(17,32,18,29,20,30)(19,31)(21,25,24,27,23,26)(22,28)$
	$ 30, 10, 3, 1 $	$44$	$30$	$( 2, 4, 3)( 5,29,21,12,13,43,18,26,38,34, 6,32,23,11,16,41,17,27,40,33, 7,30, 24,10,14,42,20,25,39,36)( 8,31,22, 9,15,44,19,28,37,35)$
	$ 10, 10, 10, 10, 1, 1, 1, 1 $	$11$	$10$	$( 5,25,14,30,40,41,23,34,18,12)( 6,26,13,29,39,42,24,33,17,11)( 7,27,16,32,38, 43,21,36,20,10)( 8,28,15,31,37,44,22,35,19, 9)$
	$ 10, 10, 10, 10, 1, 1, 1, 1 $	$11$	$10$	$( 5,30,23,12,14,41,18,25,40,34)( 6,29,24,11,13,42,17,26,39,33)( 7,32,21,10,16, 43,20,27,38,36)( 8,31,22, 9,15,44,19,28,37,35)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $	$11$	$2$	$( 5,41)( 6,42)( 7,43)( 8,44)( 9,37)(10,38)(11,39)(12,40)(13,33)(14,34)(15,35) (16,36)(17,29)(18,30)(19,31)(20,32)(21,27)(22,28)(23,25)(24,26)$
	$ 10, 10, 10, 10, 1, 1, 1, 1 $	$11$	$10$	$( 5,12,18,34,23,41,40,30,14,25)( 6,11,17,33,24,42,39,29,13,26)( 7,10,20,36,21, 43,38,32,16,27)( 8, 9,19,35,22,44,37,31,15,28)$
	$ 10, 10, 10, 10, 1, 1, 1, 1 $	$11$	$10$	$( 5,34,40,25,18,41,14,12,23,30)( 6,33,39,26,17,42,13,11,24,29)( 7,36,38,27,20, 43,16,10,21,32)( 8,35,37,28,19,44,15, 9,22,31)$
	$ 10, 10, 10, 10, 2, 2 $	$33$	$10$	$( 1,30,22,18,37,25,44, 5, 9,34)( 2,29,21,17,38,26,43, 6,10,33)( 3,32,24,20,39, 27,42, 7,11,36)( 4,31,23,19,40,28,41, 8,12,35)(13,16)(14,15)$
	$ 10, 10, 10, 10, 2, 2 $	$33$	$10$	$( 1,41,15,40,31,18,22, 5,28,34)( 2,42,16,39,32,17,21, 6,27,33)( 3,43,13,38,29, 20,24, 7,26,36)( 4,44,14,37,30,19,23, 8,25,35)( 9,12)(10,11)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$33$	$2$	$( 1,34)( 2,33)( 3,36)( 4,35)( 5,31)( 6,32)( 7,29)( 8,30)( 9,25)(10,26)(11,27) (12,28)(13,21)(14,22)(15,23)(16,24)(17,20)(18,19)(37,41)(38,42)(39,43)(40,44)$
	$ 10, 10, 10, 10, 2, 2 $	$33$	$10$	$( 1,25,31,40, 9,41,19,14, 8,34)( 2,26,32,39,10,42,20,13, 7,33)( 3,27,29,38,11, 43,17,16, 6,36)( 4,28,30,37,12,44,18,15, 5,35)(21,24)(22,23)$
	$ 10, 10, 10, 10, 2, 2 $	$33$	$10$	$( 1,12,31,14,19, 5,44,23,37,34)( 2,11,32,13,20, 6,43,24,38,33)( 3,10,29,16,17, 7,42,21,39,36)( 4, 9,30,15,18, 8,41,22,40,35)(25,28)(26,27)$
	$ 6, 6, 6, 6, 6, 3, 2, 2, 2, 2, 2, 1 $	$44$	$6$	$( 2, 3, 4)( 5,43, 6,41, 7,42)( 8,44)( 9,37)(10,39,12,38,11,40)(13,34,16,33,14, 36)(15,35)(17,30,20,29,18,32)(19,31)(21,26,23,27,24,25)(22,28)$
	$ 30, 10, 3, 1 $	$44$	$30$	$( 2, 3, 4)( 5,36,39,25,20,42,14,10,24,30, 7,33,40,27,17,41,16,11,23,32, 6,34, 38,26,18,43,13,12,21,29)( 8,35,37,28,19,44,15, 9,22,31)$
	$ 30, 10, 3, 1 $	$44$	$30$	$( 2, 3, 4)( 5,10,17,34,21,42,40,32,13,25, 7,11,18,36,24,41,38,29,14,27, 6,12, 20,33,23,43,39,30,16,26)( 8, 9,19,35,22,44,37,31,15,28)$
	$ 30, 10, 3, 1 $	$44$	$30$	$( 2, 3, 4)( 5,32,24,12,16,42,18,27,39,34, 7,29,23,10,13,41,20,26,40,36, 6,30, 21,11,14,43,17,25,38,33)( 8,31,22, 9,15,44,19,28,37,35)$
	$ 30, 10, 3, 1 $	$44$	$30$	$( 2, 3, 4)( 5,27,13,30,38,42,23,36,17,12, 7,26,14,32,39,41,21,33,18,10, 6,25, 16,29,40,43,24,34,20,11)( 8,28,15,31,37,44,22,35,19, 9)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$1320=2^{3} \cdot 3 \cdot 5 \cdot 11$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	1320.141	magma: IdentifyGroup(G);
Character table:	44 x 44 character table

magma: CharacterTable(G);

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Group invariants

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	44T46
Degree	$44$
Order	$1320$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$A_4\times F_{11}$