LMFDB - Galois group: 44T24

Show commands: Magma

magma: G := TransitiveGroup(44, 24);

Group action invariants

Degree $n$:	$44$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$24$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$C_4\times F_{11}$
Parity:	$-1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$4$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(1,4,2,3)(5,36,37,27,20,44,13,10,21,32,6,35,38,28,19,43,14,9,22,31)(7,33,39,25,18,41,15,11,23,30,8,34,40,26,17,42,16,12,24,29), (1,32,13,43,26,9,37,23,5,36,19,3,30,15,42,27,12,39,22,7,33,17,2,31,14,44,25,10,38,24,6,35,20,4,29,16,41,28,11,40,21,8,34,18)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_4$ x 2, $C_2^2$
$5$: $C_5$
$8$: $C_4\times C_2$
$10$: $C_{10}$ x 3
$20$: 20T1 x 2, 20T3
$40$: 40T2
$110$: $F_{11}$
$220$: 22T6

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$5$:	$C_5$
$8$:	$C_4\times C_2$
$10$:	$C_{10}$ x 3
$20$:	20T1 x 2, 20T3
$40$:	40T2
$110$:	$F_{11}$
$220$:	22T6

Subfields

Degree 2: $C_2$
Degree 4: $C_4$
Degree 11: $F_{11}$
Degree 22: 22T6

Low degree siblings

44T24
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	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 $	$1$	$1$	$()$
	$ 10, 10, 10, 10, 1, 1, 1, 1 $	$11$	$10$	$( 5,12,20,33,21,41,38,30,14,26)( 6,11,19,34,22,42,37,29,13,25)( 7,10,18,35,23, 43,40,31,16,27)( 8, 9,17,36,24,44,39,32,15,28)$
	$ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $	$11$	$5$	$( 5,14,38,21,20)( 6,13,37,22,19)( 7,16,40,23,18)( 8,15,39,24,17) ( 9,28,32,44,36)(10,27,31,43,35)(11,25,29,42,34)(12,26,30,41,33)$
	$ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $	$11$	$5$	$( 5,20,21,38,14)( 6,19,22,37,13)( 7,18,23,40,16)( 8,17,24,39,15) ( 9,36,44,32,28)(10,35,43,31,27)(11,34,42,29,25)(12,33,41,30,26)$
	$ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $	$11$	$5$	$( 5,21,14,20,38)( 6,22,13,19,37)( 7,23,16,18,40)( 8,24,15,17,39) ( 9,44,28,36,32)(10,43,27,35,31)(11,42,25,34,29)(12,41,26,33,30)$
	$ 10, 10, 10, 10, 1, 1, 1, 1 $	$11$	$10$	$( 5,26,14,30,38,41,21,33,20,12)( 6,25,13,29,37,42,22,34,19,11)( 7,27,16,31,40, 43,23,35,18,10)( 8,28,15,32,39,44,24,36,17, 9)$
	$ 10, 10, 10, 10, 1, 1, 1, 1 $	$11$	$10$	$( 5,30,21,12,14,41,20,26,38,33)( 6,29,22,11,13,42,19,25,37,34)( 7,31,23,10,16, 43,18,27,40,35)( 8,32,24, 9,15,44,17,28,39,36)$
	$ 10, 10, 10, 10, 1, 1, 1, 1 $	$11$	$10$	$( 5,33,38,26,20,41,14,12,21,30)( 6,34,37,25,19,42,13,11,22,29)( 7,35,40,27,18, 43,16,10,23,31)( 8,36,39,28,17,44,15, 9,24,32)$
	$ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $	$11$	$5$	$( 5,38,20,14,21)( 6,37,19,13,22)( 7,40,18,16,23)( 8,39,17,15,24) ( 9,32,36,28,44)(10,31,35,27,43)(11,29,34,25,42)(12,30,33,26,41)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $	$11$	$2$	$( 5,41)( 6,42)( 7,43)( 8,44)( 9,39)(10,40)(11,37)(12,38)(13,34)(14,33)(15,36) (16,35)(17,32)(18,31)(19,29)(20,30)(21,26)(22,25)(23,27)(24,28)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)$
	$ 10, 10, 10, 10, 2, 2 $	$11$	$10$	$( 1, 2)( 3, 4)( 5,11,20,34,21,42,38,29,14,25)( 6,12,19,33,22,41,37,30,13,26) ( 7, 9,18,36,23,44,40,32,16,28)( 8,10,17,35,24,43,39,31,15,27)$
	$ 10, 10, 10, 10, 2, 2 $	$11$	$10$	$( 1, 2)( 3, 4)( 5,13,38,22,20, 6,14,37,21,19)( 7,15,40,24,18, 8,16,39,23,17) ( 9,27,32,43,36,10,28,31,44,35)(11,26,29,41,34,12,25,30,42,33)$
	$ 10, 10, 10, 10, 2, 2 $	$11$	$10$	$( 1, 2)( 3, 4)( 5,19,21,37,14, 6,20,22,38,13)( 7,17,23,39,16, 8,18,24,40,15) ( 9,35,44,31,28,10,36,43,32,27)(11,33,42,30,25,12,34,41,29,26)$
	$ 10, 10, 10, 10, 2, 2 $	$11$	$10$	$( 1, 2)( 3, 4)( 5,22,14,19,38, 6,21,13,20,37)( 7,24,16,17,40, 8,23,15,18,39) ( 9,43,28,35,32,10,44,27,36,31)(11,41,25,33,29,12,42,26,34,30)$
	$ 10, 10, 10, 10, 2, 2 $	$11$	$10$	$( 1, 2)( 3, 4)( 5,25,14,29,38,42,21,34,20,11)( 6,26,13,30,37,41,22,33,19,12) ( 7,28,16,32,40,44,23,36,18, 9)( 8,27,15,31,39,43,24,35,17,10)$
	$ 10, 10, 10, 10, 2, 2 $	$11$	$10$	$( 1, 2)( 3, 4)( 5,29,21,11,14,42,20,25,38,34)( 6,30,22,12,13,41,19,26,37,33) ( 7,32,23, 9,16,44,18,28,40,36)( 8,31,24,10,15,43,17,27,39,35)$
	$ 10, 10, 10, 10, 2, 2 $	$11$	$10$	$( 1, 2)( 3, 4)( 5,34,38,25,20,42,14,11,21,29)( 6,33,37,26,19,41,13,12,22,30) ( 7,36,40,28,18,44,16, 9,23,32)( 8,35,39,27,17,43,15,10,24,31)$
	$ 10, 10, 10, 10, 2, 2 $	$11$	$10$	$( 1, 2)( 3, 4)( 5,37,20,13,21, 6,38,19,14,22)( 7,39,18,15,23, 8,40,17,16,24) ( 9,31,36,27,44,10,32,35,28,43)(11,30,34,26,42,12,29,33,25,41)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$11$	$2$	$( 1, 2)( 3, 4)( 5,42)( 6,41)( 7,44)( 8,43)( 9,40)(10,39)(11,38)(12,37)(13,33) (14,34)(15,35)(16,36)(17,31)(18,32)(19,30)(20,29)(21,25)(22,26)(23,28)(24,27)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)(17,20,18,19)(21,23,22,24) (25,28,26,27)(29,32,30,31)(33,35,34,36)(37,39,38,40)(41,43,42,44)$
	$ 20, 20, 4 $	$11$	$20$	$( 1, 3, 2, 4)( 5,10,19,36,21,43,37,32,14,27, 6, 9,20,35,22,44,38,31,13,28) ( 7,11,17,33,23,42,39,30,16,25, 8,12,18,34,24,41,40,29,15,26)$
	$ 20, 20, 4 $	$11$	$20$	$( 1, 3, 2, 4)( 5,16,37,24,20, 7,13,39,21,18, 6,15,38,23,19, 8,14,40,22,17) ( 9,26,31,42,36,12,27,29,44,33,10,25,32,41,35,11,28,30,43,34)$
	$ 20, 20, 4 $	$11$	$20$	$( 1, 3, 2, 4)( 5,18,22,39,14, 7,19,24,38,16, 6,17,21,40,13, 8,20,23,37,15) ( 9,33,43,29,28,12,35,42,32,26,10,34,44,30,27,11,36,41,31,25)$
	$ 20, 20, 4 $	$11$	$20$	$( 1, 3, 2, 4)( 5,23,13,17,38, 7,22,15,20,40, 6,24,14,18,37, 8,21,16,19,39) ( 9,41,27,34,32,12,43,25,36,30,10,42,28,33,31,11,44,26,35,29)$
	$ 20, 20, 4 $	$11$	$20$	$( 1, 3, 2, 4)( 5,27,13,32,38,43,22,36,20,10, 6,28,14,31,37,44,21,35,19, 9) ( 7,25,15,30,40,42,24,33,18,11, 8,26,16,29,39,41,23,34,17,12)$
	$ 20, 20, 4 $	$11$	$20$	$( 1, 3, 2, 4)( 5,31,22, 9,14,43,19,28,38,35, 6,32,21,10,13,44,20,27,37,36) ( 7,29,24,12,16,42,17,26,40,34, 8,30,23,11,15,41,18,25,39,33)$
	$ 20, 20, 4 $	$11$	$20$	$( 1, 3, 2, 4)( 5,35,37,28,20,43,13, 9,21,31, 6,36,38,27,19,44,14,10,22,32) ( 7,34,39,26,18,42,15,12,23,29, 8,33,40,25,17,41,16,11,24,30)$
	$ 20, 20, 4 $	$11$	$20$	$( 1, 3, 2, 4)( 5,40,19,15,21, 7,37,17,14,23, 6,39,20,16,22, 8,38,18,13,24) ( 9,30,35,25,44,12,31,34,28,41,10,29,36,26,43,11,32,33,27,42)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $	$11$	$4$	$( 1, 3, 2, 4)( 5,43, 6,44)( 7,42, 8,41)( 9,38,10,37)(11,39,12,40)(13,36,14,35) (15,33,16,34)(17,30,18,29)(19,32,20,31)(21,27,22,28)(23,25,24,26)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)(17,19,18,20)(21,24,22,23) (25,27,26,28)(29,31,30,32)(33,36,34,35)(37,40,38,39)(41,44,42,43)$
	$ 20, 20, 4 $	$11$	$20$	$( 1, 4, 2, 3)( 5, 9,19,35,21,44,37,31,14,28, 6,10,20,36,22,43,38,32,13,27) ( 7,12,17,34,23,41,39,29,16,26, 8,11,18,33,24,42,40,30,15,25)$
	$ 20, 20, 4 $	$11$	$20$	$( 1, 4, 2, 3)( 5,15,37,23,20, 8,13,40,21,17, 6,16,38,24,19, 7,14,39,22,18) ( 9,25,31,41,36,11,27,30,44,34,10,26,32,42,35,12,28,29,43,33)$
	$ 20, 20, 4 $	$11$	$20$	$( 1, 4, 2, 3)( 5,17,22,40,14, 8,19,23,38,15, 6,18,21,39,13, 7,20,24,37,16) ( 9,34,43,30,28,11,35,41,32,25,10,33,44,29,27,12,36,42,31,26)$
	$ 20, 20, 4 $	$11$	$20$	$( 1, 4, 2, 3)( 5,24,13,18,38, 8,22,16,20,39, 6,23,14,17,37, 7,21,15,19,40) ( 9,42,27,33,32,11,43,26,36,29,10,41,28,34,31,12,44,25,35,30)$
	$ 20, 20, 4 $	$11$	$20$	$( 1, 4, 2, 3)( 5,28,13,31,38,44,22,35,20, 9, 6,27,14,32,37,43,21,36,19,10) ( 7,26,15,29,40,41,24,34,18,12, 8,25,16,30,39,42,23,33,17,11)$
	$ 20, 20, 4 $	$11$	$20$	$( 1, 4, 2, 3)( 5,32,22,10,14,44,19,27,38,36, 6,31,21, 9,13,43,20,28,37,35) ( 7,30,24,11,16,41,17,25,40,33, 8,29,23,12,15,42,18,26,39,34)$
	$ 20, 20, 4 $	$11$	$20$	$( 1, 4, 2, 3)( 5,36,37,27,20,44,13,10,21,32, 6,35,38,28,19,43,14, 9,22,31) ( 7,33,39,25,18,41,15,11,23,30, 8,34,40,26,17,42,16,12,24,29)$
	$ 20, 20, 4 $	$11$	$20$	$( 1, 4, 2, 3)( 5,39,19,16,21, 8,37,18,14,24, 6,40,20,15,22, 7,38,17,13,23) ( 9,29,35,26,44,11,31,33,28,42,10,30,36,25,43,12,32,34,27,41)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $	$11$	$4$	$( 1, 4, 2, 3)( 5,44, 6,43)( 7,41, 8,42)( 9,37,10,38)(11,40,12,39)(13,35,14,36) (15,34,16,33)(17,29,18,30)(19,31,20,32)(21,28,22,27)(23,26,24,25)$
	$ 11, 11, 11, 11 $	$10$	$11$	$( 1, 5,12,14,20,21,26,30,33,38,41)( 2, 6,11,13,19,22,25,29,34,37,42) ( 3, 7,10,16,18,23,27,31,35,40,43)( 4, 8, 9,15,17,24,28,32,36,39,44)$
	$ 22, 22 $	$10$	$22$	$( 1, 6,12,13,20,22,26,29,33,37,41, 2, 5,11,14,19,21,25,30,34,38,42) ( 3, 8,10,15,18,24,27,32,35,39,43, 4, 7, 9,16,17,23,28,31,36,40,44)$
	$ 44 $	$10$	$44$	$( 1, 7,11,15,20,23,25,32,33,40,42, 4, 5,10,13,17,21,27,29,36,38,43, 2, 8,12, 16,19,24,26,31,34,39,41, 3, 6, 9,14,18,22,28,30,35,37,44)$
	$ 44 $	$10$	$44$	$( 1, 8,11,16,20,24,25,31,33,39,42, 3, 5, 9,13,18,21,28,29,35,38,44, 2, 7,12, 15,19,23,26,32,34,40,41, 4, 6,10,14,17,22,27,30,36,37,43)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$440=2^{3} \cdot 5 \cdot 11$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	440.8	magma: IdentifyGroup(G);
Character table:	44 x 44 character table

magma: CharacterTable(G);

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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	44T24
Degree	$44$
Order	$440$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_4\times F_{11}$