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Properties

Label 44T19
Degree $44$
Order $264$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{11}:S_4$

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Show commands: Magma

magma: G := TransitiveGroup(44, 19);
 

Group action invariants

Degree $n$:  $44$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $19$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{11}:S_4$
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:  (1,41,39,35,29,27,24,19,16,12,6,3,42,38,33,30,26,22,20,13,10,5,2,44,37,36,32,25,23,18,14,11,7)(4,43,40,34,31,28,21,17,15,9,8), (1,26,4,27)(2,28,3,25)(5,23,8,22)(6,21,7,24)(9,18,12,19)(10,20,11,17)(13,15,16,14)(29,43,32,42)(30,41,31,44)(33,37,36,40)(34,39,35,38)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$22$:  $D_{11}$
$24$:  $S_4$
$66$:  $D_{33}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $S_4$

Degree 11: $D_{11}$

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

LabelCycle TypeSizeOrderRepresentative
$ 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $66$ $2$ $( 3, 4)( 5,42)( 6,41)( 7,43)( 8,44)( 9,39)(10,40)(11,38)(12,37)(13,36)(14,35) (15,33)(16,34)(17,32)(18,31)(19,29)(20,30)(21,27)(22,28)(23,26)(24,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 2, 3, 4)( 6, 7, 8)( 9,11,10)(13,16,15)(17,19,18)(21,23,22)(26,27,28) (29,32,31)(33,34,36)(38,39,40)(41,44,43)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $66$ $4$ $( 1, 2, 3, 4)( 5,41, 7,43)( 6,44, 8,42)( 9,37,11,39)(10,40,12,38)(13,33,15,35) (14,36,16,34)(17,30,19,32)(18,31,20,29)(21,25,23,27)(22,28,24,26)$
$ 11, 11, 11, 11 $ $2$ $11$ $( 1, 5,12,14,20,24,25,30,35,37,42)( 2, 6,11,13,19,23,26,29,36,38,41) ( 3, 7,10,16,18,22,27,32,33,39,44)( 4, 8, 9,15,17,21,28,31,34,40,43)$
$ 33, 11 $ $8$ $33$ $( 1, 5,12,14,20,24,25,30,35,37,42)( 2, 7, 9,13,18,21,26,32,34,38,44, 4, 6,10, 15,19,22,28,29,33,40,41, 3, 8,11,16,17,23,27,31,36,39,43)$
$ 33, 11 $ $8$ $33$ $( 1, 5,12,14,20,24,25,30,35,37,42)( 2, 8,10,13,17,22,26,31,33,38,43, 3, 6, 9, 16,19,21,27,29,34,39,41, 4, 7,11,15,18,23,28,32,36,40,44)$
$ 22, 22 $ $6$ $22$ $( 1, 6,12,13,20,23,25,29,35,38,42, 2, 5,11,14,19,24,26,30,36,37,41) ( 3, 8,10,15,18,21,27,31,33,40,44, 4, 7, 9,16,17,22,28,32,34,39,43)$
$ 22, 22 $ $6$ $22$ $( 1, 9,20,28,35,43, 5,15,24,31,37, 4,12,17,25,34,42, 8,14,21,30,40) ( 2,10,19,27,36,44, 6,16,23,32,38, 3,11,18,26,33,41, 7,13,22,29,39)$
$ 33, 11 $ $8$ $33$ $( 1, 9,18,25,34,44, 5,15,22,30,40, 3,12,17,27,35,43, 7,14,21,32,37, 4,10,20, 28,33,42, 8,16,24,31,39)( 2,11,19,26,36,41, 6,13,23,29,38)$
$ 33, 11 $ $8$ $33$ $( 1, 9,19,25,34,41, 5,15,23,30,40, 2,12,17,26,35,43, 6,14,21,29,37, 4,11,20, 28,36,42, 8,13,24,31,38)( 3,10,18,27,33,44, 7,16,22,32,39)$
$ 11, 11, 11, 11 $ $2$ $11$ $( 1,12,20,25,35,42, 5,14,24,30,37)( 2,11,19,26,36,41, 6,13,23,29,38) ( 3,10,18,27,33,44, 7,16,22,32,39)( 4, 9,17,28,34,43, 8,15,21,31,40)$
$ 22, 22 $ $6$ $22$ $( 1,13,25,38, 5,19,30,41,12,23,35, 2,14,26,37, 6,20,29,42,11,24,36) ( 3,15,27,40, 7,17,32,43,10,21,33, 4,16,28,39, 8,18,31,44, 9,22,34)$
$ 33, 11 $ $8$ $33$ $( 1,13,28,37, 6,17,30,41, 9,24,36, 4,14,26,40, 5,19,31,42,11,21,35, 2,15,25, 38, 8,20,29,43,12,23,34)( 3,16,27,39, 7,18,32,44,10,22,33)$
$ 33, 11 $ $8$ $33$ $( 1,13,27,37, 6,18,30,41,10,24,36, 3,14,26,39, 5,19,32,42,11,22,35, 2,16,25, 38, 7,20,29,44,12,23,33)( 4,15,28,40, 8,17,31,43, 9,21,34)$
$ 11, 11, 11, 11 $ $2$ $11$ $( 1,14,25,37, 5,20,30,42,12,24,35)( 2,13,26,38, 6,19,29,41,11,23,36) ( 3,16,27,39, 7,18,32,44,10,22,33)( 4,15,28,40, 8,17,31,43, 9,21,34)$
$ 22, 22 $ $6$ $22$ $( 1,17,35, 8,24,40,12,28,42,15,30, 4,20,34, 5,21,37, 9,25,43,14,31) ( 2,18,36, 7,23,39,11,27,41,16,29, 3,19,33, 6,22,38,10,26,44,13,32)$
$ 33, 11 $ $8$ $33$ $( 1,17,33, 5,21,39,12,28,44,14,31, 3,20,34, 7,24,40,10,25,43,16,30, 4,18,35, 8,22,37, 9,27,42,15,32)( 2,19,36, 6,23,38,11,26,41,13,29)$
$ 33, 11 $ $8$ $33$ $( 1,17,36, 5,21,38,12,28,41,14,31, 2,20,34, 6,24,40,11,25,43,13,30, 4,19,35, 8,23,37, 9,26,42,15,29)( 3,18,33, 7,22,39,10,27,44,16,32)$
$ 11, 11, 11, 11 $ $2$ $11$ $( 1,20,35, 5,24,37,12,25,42,14,30)( 2,19,36, 6,23,38,11,26,41,13,29) ( 3,18,33, 7,22,39,10,27,44,16,32)( 4,17,34, 8,21,40, 9,28,43,15,31)$
$ 22, 22 $ $6$ $22$ $( 1,21,42,17,37,15,35, 9,30, 8,25, 4,24,43,20,40,14,34,12,31, 5,28) ( 2,22,41,18,38,16,36,10,29, 7,26, 3,23,44,19,39,13,33,11,32, 6,27)$
$ 33, 11 $ $8$ $33$ $( 1,21,44,20,40,16,35, 9,32, 5,28, 3,24,43,18,37,15,33,12,31, 7,25, 4,22,42, 17,39,14,34,10,30, 8,27)( 2,23,41,19,38,13,36,11,29, 6,26)$
$ 33, 11 $ $8$ $33$ $( 1,21,41,20,40,13,35, 9,29, 5,28, 2,24,43,19,37,15,36,12,31, 6,25, 4,23,42, 17,38,14,34,11,30, 8,26)( 3,22,44,18,39,16,33,10,32, 7,27)$
$ 11, 11, 11, 11 $ $2$ $11$ $( 1,24,42,20,37,14,35,12,30, 5,25)( 2,23,41,19,38,13,36,11,29, 6,26) ( 3,22,44,18,39,16,33,10,32, 7,27)( 4,21,43,17,40,15,34, 9,31, 8,28)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $264=2^{3} \cdot 3 \cdot 11$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  264.32
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A 4A 11A1 11A2 11A3 11A4 11A5 22A1 22A3 22A5 22A7 22A9 33A1 33A2 33A4 33A5 33A7 33A8 33A10 33A13 33A14 33A16
Size 1 3 66 8 66 2 2 2 2 2 6 6 6 6 6 8 8 8 8 8 8 8 8 8 8
2 P 1A 1A 1A 3A 2A 11A4 11A2 11A1 11A3 11A5 11A4 11A5 11A1 11A2 11A3 33A5 33A1 33A7 33A10 33A14 33A2 33A8 33A4 33A16 33A13
3 P 1A 2A 2B 1A 4A 11A5 11A3 11A4 11A1 11A2 22A7 22A5 22A1 22A9 22A3 11A3 11A5 11A2 11A5 11A4 11A1 11A4 11A2 11A3 11A1
11 P 1A 2A 2B 3A 4A 11A1 11A5 11A3 11A2 11A4 22A3 22A1 22A9 22A7 22A5 33A4 33A14 33A1 33A8 33A2 33A5 33A13 33A10 33A7 33A16
Type
264.32.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
264.32.1b R 1 1 −1 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
264.32.2a R 2 2 0 −1 0 2 2 2 2 2 2 2 2 2 2 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1
264.32.2b1 R 2 2 0 2 0 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−4+ζ114 ζ11−2+ζ112 ζ11−3+ζ113 ζ11−2+ζ112 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−3+ζ113 ζ11−4+ζ114 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−1+ζ11 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−5+ζ115 ζ11−1+ζ11
264.32.2b2 R 2 2 0 2 0 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−1+ζ11 ζ11−5+ζ115 ζ11−2+ζ112 ζ11−5+ζ115 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−2+ζ112 ζ11−1+ζ11 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−3+ζ113 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−4+ζ114 ζ11−3+ζ113
264.32.2b3 R 2 2 0 2 0 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−2+ζ112 ζ11−1+ζ11 ζ11−4+ζ114 ζ11−1+ζ11 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−4+ζ114 ζ11−2+ζ112 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−5+ζ115 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−3+ζ113 ζ11−5+ζ115
264.32.2b4 R 2 2 0 2 0 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−5+ζ115 ζ11−3+ζ113 ζ11−1+ζ11 ζ11−3+ζ113 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−1+ζ11 ζ11−5+ζ115 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−4+ζ114 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−2+ζ112 ζ11−4+ζ114
264.32.2b5 R 2 2 0 2 0 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−3+ζ113 ζ11−4+ζ114 ζ11−5+ζ115 ζ11−4+ζ114 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−5+ζ115 ζ11−3+ζ113 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−2+ζ112 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−1+ζ11 ζ11−2+ζ112
264.32.2c1 R 2 2 0 −1 0 ζ33−15+ζ3315 ζ33−3+ζ333 ζ33−12+ζ3312 ζ33−6+ζ336 ζ33−9+ζ339 ζ33−6+ζ336 ζ33−15+ζ3315 ζ33−3+ζ333 ζ33−9+ζ339 ζ33−12+ζ3312 ζ33−16+ζ3316 ζ33−1+ζ33 ζ33−2+ζ332 ζ33−14+ζ3314 ζ33−13+ζ3313 ζ33−4+ζ334 ζ33−5+ζ335 ζ33−10+ζ3310 ζ33−7+ζ337 ζ33−8+ζ338
264.32.2c2 R 2 2 0 −1 0 ζ33−15+ζ3315 ζ33−3+ζ333 ζ33−12+ζ3312 ζ33−6+ζ336 ζ33−9+ζ339 ζ33−6+ζ336 ζ33−15+ζ3315 ζ33−3+ζ333 ζ33−9+ζ339 ζ33−12+ζ3312 ζ33−5+ζ335 ζ33−10+ζ3310 ζ33−13+ζ3313 ζ33−8+ζ338 ζ33−2+ζ332 ζ33−7+ζ337 ζ33−16+ζ3316 ζ33−1+ζ33 ζ33−4+ζ334 ζ33−14+ζ3314
264.32.2c3 R 2 2 0 −1 0 ζ33−12+ζ3312 ζ33−9+ζ339 ζ33−3+ζ333 ζ33−15+ζ3315 ζ33−6+ζ336 ζ33−15+ζ3315 ζ33−12+ζ3312 ζ33−9+ζ339 ζ33−6+ζ336 ζ33−3+ζ333 ζ33−7+ζ337 ζ33−14+ζ3314 ζ33−5+ζ335 ζ33−2+ζ332 ζ33−16+ζ3316 ζ33−10+ζ3310 ζ33−4+ζ334 ζ33−8+ζ338 ζ33−1+ζ33 ζ33−13+ζ3313
264.32.2c4 R 2 2 0 −1 0 ζ33−12+ζ3312 ζ33−9+ζ339 ζ33−3+ζ333 ζ33−15+ζ3315 ζ33−6+ζ336 ζ33−15+ζ3315 ζ33−12+ζ3312 ζ33−9+ζ339 ζ33−6+ζ336 ζ33−3+ζ333 ζ33−4+ζ334 ζ33−8+ζ338 ζ33−16+ζ3316 ζ33−13+ζ3313 ζ33−5+ζ335 ζ33−1+ζ33 ζ33−7+ζ337 ζ33−14+ζ3314 ζ33−10+ζ3310 ζ33−2+ζ332
264.32.2c5 R 2 2 0 −1 0 ζ33−9+ζ339 ζ33−15+ζ3315 ζ33−6+ζ336 ζ33−3+ζ333 ζ33−12+ζ3312 ζ33−3+ζ333 ζ33−9+ζ339 ζ33−15+ζ3315 ζ33−12+ζ3312 ζ33−6+ζ336 ζ33−14+ζ3314 ζ33−5+ζ335 ζ33−10+ζ3310 ζ33−4+ζ334 ζ33−1+ζ33 ζ33−13+ζ3313 ζ33−8+ζ338 ζ33−16+ζ3316 ζ33−2+ζ332 ζ33−7+ζ337
264.32.2c6 R 2 2 0 −1 0 ζ33−9+ζ339 ζ33−15+ζ3315 ζ33−6+ζ336 ζ33−3+ζ333 ζ33−12+ζ3312 ζ33−3+ζ333 ζ33−9+ζ339 ζ33−15+ζ3315 ζ33−12+ζ3312 ζ33−6+ζ336 ζ33−8+ζ338 ζ33−16+ζ3316 ζ33−1+ζ33 ζ33−7+ζ337 ζ33−10+ζ3310 ζ33−2+ζ332 ζ33−14+ζ3314 ζ33−5+ζ335 ζ33−13+ζ3313 ζ33−4+ζ334
264.32.2c7 R 2 2 0 −1 0 ζ33−6+ζ336 ζ33−12+ζ3312 ζ33−15+ζ3315 ζ33−9+ζ339 ζ33−3+ζ333 ζ33−9+ζ339 ζ33−6+ζ336 ζ33−12+ζ3312 ζ33−3+ζ333 ζ33−15+ζ3315 ζ33−13+ζ3313 ζ33−7+ζ337 ζ33−14+ζ3314 ζ33−1+ζ33 ζ33−8+ζ338 ζ33−5+ζ335 ζ33−2+ζ332 ζ33−4+ζ334 ζ33−16+ζ3316 ζ33−10+ζ3310
264.32.2c8 R 2 2 0 −1 0 ζ33−6+ζ336 ζ33−12+ζ3312 ζ33−15+ζ3315 ζ33−9+ζ339 ζ33−3+ζ333 ζ33−9+ζ339 ζ33−6+ζ336 ζ33−12+ζ3312 ζ33−3+ζ333 ζ33−15+ζ3315 ζ33−2+ζ332 ζ33−4+ζ334 ζ33−8+ζ338 ζ33−10+ζ3310 ζ33−14+ζ3314 ζ33−16+ζ3316 ζ33−13+ζ3313 ζ33−7+ζ337 ζ33−5+ζ335 ζ33−1+ζ33
264.32.2c9 R 2 2 0 −1 0 ζ33−3+ζ333 ζ33−6+ζ336 ζ33−9+ζ339 ζ33−12+ζ3312 ζ33−15+ζ3315 ζ33−12+ζ3312 ζ33−3+ζ333 ζ33−6+ζ336 ζ33−15+ζ3315 ζ33−9+ζ339 ζ33−10+ζ3310 ζ33−13+ζ3313 ζ33−7+ζ337 ζ33−16+ζ3316 ζ33−4+ζ334 ζ33−14+ζ3314 ζ33−1+ζ33 ζ33−2+ζ332 ζ33−8+ζ338 ζ33−5+ζ335
264.32.2c10 R 2 2 0 −1 0 ζ33−3+ζ333 ζ33−6+ζ336 ζ33−9+ζ339 ζ33−12+ζ3312 ζ33−15+ζ3315 ζ33−12+ζ3312 ζ33−3+ζ333 ζ33−6+ζ336 ζ33−15+ζ3315 ζ33−9+ζ339 ζ33−1+ζ33 ζ33−2+ζ332 ζ33−4+ζ334 ζ33−5+ζ335 ζ33−7+ζ337 ζ33−8+ζ338 ζ33−10+ζ3310 ζ33−13+ζ3313 ζ33−14+ζ3314 ζ33−16+ζ3316
264.32.3a R 3 −1 1 0 −1 3 3 3 3 3 −1 −1 −1 −1 −1 0 0 0 0 0 0 0 0 0 0
264.32.3b R 3 −1 −1 0 1 3 3 3 3 3 −1 −1 −1 −1 −1 0 0 0 0 0 0 0 0 0 0
264.32.6a1 R 6 −2 0 0 0 3ζ11−5+3ζ115 3ζ11−1+3ζ11 3ζ11−4+3ζ114 3ζ11−2+3ζ112 3ζ11−3+3ζ113 −ζ11−2−ζ112 −ζ11−5−ζ115 −ζ11−1−ζ11 −ζ11−3−ζ113 −ζ11−4−ζ114 0 0 0 0 0 0 0 0 0 0
264.32.6a2 R 6 −2 0 0 0 3ζ11−4+3ζ114 3ζ11−3+3ζ113 3ζ11−1+3ζ11 3ζ11−5+3ζ115 3ζ11−2+3ζ112 −ζ11−5−ζ115 −ζ11−4−ζ114 −ζ11−3−ζ113 −ζ11−2−ζ112 −ζ11−1−ζ11 0 0 0 0 0 0 0 0 0 0
264.32.6a3 R 6 −2 0 0 0 3ζ11−3+3ζ113 3ζ11−5+3ζ115 3ζ11−2+3ζ112 3ζ11−1+3ζ11 3ζ11−4+3ζ114 −ζ11−1−ζ11 −ζ11−3−ζ113 −ζ11−5−ζ115 −ζ11−4−ζ114 −ζ11−2−ζ112 0 0 0 0 0 0 0 0 0 0
264.32.6a4 R 6 −2 0 0 0 3ζ11−2+3ζ112 3ζ11−4+3ζ114 3ζ11−5+3ζ115 3ζ11−3+3ζ113 3ζ11−1+3ζ11 −ζ11−3−ζ113 −ζ11−2−ζ112 −ζ11−4−ζ114 −ζ11−1−ζ11 −ζ11−5−ζ115 0 0 0 0 0 0 0 0 0 0
264.32.6a5 R 6 −2 0 0 0 3ζ11−1+3ζ11 3ζ11−2+3ζ112 3ζ11−3+3ζ113 3ζ11−4+3ζ114 3ζ11−5+3ζ115 −ζ11−4−ζ114 −ζ11−1−ζ11 −ζ11−2−ζ112 −ζ11−5−ζ115 −ζ11−3−ζ113 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);