Properties

Label 44T43
Order \(968\)
n \(44\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

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Group action invariants

Degree $n$ :  $44$
Transitive number $t$ :  $43$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,34,7,43)(2,41,6,36)(3,37,5,40)(4,44)(8,39,11,38)(9,35,10,42)(12,29,19,27)(13,24,18,32)(14,30,17,26)(15,25,16,31)(20,33,22,23)(21,28), (1,33,8,32)(2,25,7,29)(3,28,6,26)(4,31,5,23)(9,24,11,30)(10,27)(12,36,14,41)(13,44)(15,38,22,39)(16,35,21,42)(17,43,20,34)(18,40,19,37)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $Q_8$

Resolvents shown for degrees $\leq 29$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 11: None

Degree 22: None

Low degree siblings

There are no siblings with degree $\leq 29$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

Conjugacy Classes

Cycle 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 $ $1$ $1$ $()$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,22,21,20,19,18,17,16,15,14,13) (23,33,32,31,30,29,28,27,26,25,24)(34,42,39,36,44,41,38,35,43,40,37)$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1, 7, 2, 8, 3, 9, 4,10, 5,11, 6)(12,21,19,17,15,13,22,20,18,16,14) (23,32,30,28,26,24,33,31,29,27,25)(34,39,44,38,43,37,42,36,41,35,40)$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,19,15,22,18,14,21,17,13,20,16) (23,30,26,33,29,25,32,28,24,31,27)(34,44,43,42,41,40,39,38,37,36,35)$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,15,18,21,13,16,19,22,14,17,20) (23,26,29,32,24,27,30,33,25,28,31)(34,43,41,39,37,35,44,42,40,38,36)$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1, 5, 9, 2, 6,10, 3, 7,11, 4, 8)(12,18,13,19,14,20,15,21,16,22,17) (23,29,24,30,25,31,26,32,27,33,28)(34,41,37,44,40,36,43,39,35,42,38)$
$ 11, 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $11$ $(12,22,21,20,19,18,17,16,15,14,13)(23,25,27,29,31,33,24,26,28,30,32) (34,35,36,37,38,39,40,41,42,43,44)$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,21,19,17,15,13,22,20,18,16,14) (23,24,25,26,27,28,29,30,31,32,33)(34,43,41,39,37,35,44,42,40,38,36)$
$ 11, 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $11$ $( 1, 7, 2, 8, 3, 9, 4,10, 5,11, 6)(12,20,17,14,22,19,16,13,21,18,15) (34,40,35,41,36,42,37,43,38,44,39)$
$ 11, 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,18,13,19,14,20,15,21,16,22,17) (23,32,30,28,26,24,33,31,29,27,25)$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1, 5, 9, 2, 6,10, 3, 7,11, 4, 8)(12,17,22,16,21,15,20,14,19,13,18) (23,31,28,25,33,30,27,24,32,29,26)(34,42,39,36,44,41,38,35,43,40,37)$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,19,15,22,18,14,21,17,13,20,16) (23,33,32,31,30,29,28,27,26,25,24)(34,37,40,43,35,38,41,44,36,39,42)$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,20,17,14,22,19,16,13,21,18,15) (23,26,29,32,24,27,30,33,25,28,31)(34,44,43,42,41,40,39,38,37,36,35)$
$ 11, 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $11$ $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,13,14,15,16,17,18,19,20,21,22) (23,30,26,33,29,25,32,28,24,31,27)$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1, 7, 2, 8, 3, 9, 4,10, 5,11, 6)(12,17,22,16,21,15,20,14,19,13,18) (23,29,24,30,25,31,26,32,27,33,28)(34,43,41,39,37,35,44,42,40,38,36)$
$ 11, 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $11$ $( 1, 5, 9, 2, 6,10, 3, 7,11, 4, 8)(12,14,16,18,20,22,13,15,17,19,21) (23,26,29,32,24,27,30,33,25,28,31)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $121$ $2$ $( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,22)(14,21)(15,20)(16,19)(17,18)(23,31) (24,30)(25,29)(26,28)(32,33)(34,36)(37,44)(38,43)(39,42)(40,41)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2 $ $242$ $4$ $( 1,34, 7,43)( 2,41, 6,36)( 3,37, 5,40)( 4,44)( 8,39,11,38)( 9,35,10,42) (12,29,19,27)(13,24,18,32)(14,30,17,26)(15,25,16,31)(20,33,22,23)(21,28)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2 $ $242$ $4$ $( 1,33, 8,32)( 2,25, 7,29)( 3,28, 6,26)( 4,31, 5,23)( 9,24,11,30)(10,27) (12,36,14,41)(13,44)(15,38,22,39)(16,35,21,42)(17,43,20,34)(18,40,19,37)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2 $ $242$ $4$ $( 1,22,11,17)( 2,16,10,12)( 3,21, 9,18)( 4,15, 8,13)( 5,20, 7,19)( 6,14) (23,44,24,38)(25,43,33,39)(26,37,32,34)(27,42,31,40)(28,36,30,35)(29,41)$

Group invariants

Order:  $968=2^{3} \cdot 11^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [968, 37]
Character table:   
      2  3   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .  3  2
     11  2   2   2   2   2   2   2   2   2   2   2   2   2   2   2   2  .  .

        1a 11a 11b 11c 11d 11e 11f 11g 11h 11i 11j 11k 11l 11m 11n 11o 2a 4a
     2P 1a 11b 11c 11d 11e 11a 11h 11l 11i 11m 11k 11g 11n 11o 11j 11f 1a 2a
     3P 1a 11d 11e 11a 11b 11c 11m 11j 11o 11f 11l 11n 11k 11h 11g 11i 2a 4a
     5P 1a 11e 11a 11b 11c 11d 11o 11k 11f 11h 11n 11j 11g 11i 11l 11m 2a 4a
     7P 1a 11c 11d 11e 11a 11b 11i 11n 11m 11o 11g 11l 11j 11f 11k 11h 2a 4a
    11P 1a  1a  1a  1a  1a  1a  1a  1a  1a  1a  1a  1a  1a  1a  1a  1a 2a 4a

X.1      1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1  1  1
X.2      1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1  1 -1
X.3      1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1  1 -1
X.4      1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1  1  1
X.5      2   2   2   2   2   2   2   2   2   2   2   2   2   2   2   2 -2  .
X.6      8   A   C   D   B   E   J   F   G   F   J   G   H   H   I   I  .  .
X.7      8   B   E   A   C   D   H   J   I   J   H   I   G   G   F   F  .  .
X.8      8   C   D   B   E   A   G   H   F   H   G   F   I   I   J   J  .  .
X.9      8   D   B   E   A   C   F   I   H   I   F   H   J   J   G   G  .  .
X.10     8   E   A   C   D   B   I   G   J   G   I   J   F   F   H   H  .  .
X.11     8   F   H   I   J   G   A   H   C   D   G   F   I   B   J   E  .  .
X.12     8   G   F   H   I   J   E   F   A   C   J   G   H   D   I   B  .  .
X.13     8   H   I   J   G   F   C   I   D   B   F   H   J   E   G   A  .  .
X.14     8   I   J   G   F   H   D   J   B   E   H   I   G   A   F   C  .  .
X.15     8   J   G   F   H   I   B   G   E   A   I   J   F   C   H   D  .  .
X.16     8   F   H   I   J   G   G   D   F   H   A   C   B   I   E   J  .  .
X.17     8   G   F   H   I   J   J   C   G   F   E   A   D   H   B   I  .  .
X.18     8   H   I   J   G   F   F   B   H   I   C   D   E   J   A   G  .  .
X.19     8   I   J   G   F   H   H   E   I   J   D   B   A   G   C   F  .  .
X.20     8   J   G   F   H   I   I   A   J   G   B   E   C   F   D   H  .  .

      2  2  2
     11  .  .

        4b 4c
     2P 2a 2a
     3P 4b 4c
     5P 4b 4c
     7P 4b 4c
    11P 4b 4c

X.1      1  1
X.2     -1  1
X.3      1 -1
X.4     -1 -1
X.5      .  .
X.6      .  .
X.7      .  .
X.8      .  .
X.9      .  .
X.10     .  .
X.11     .  .
X.12     .  .
X.13     .  .
X.14     .  .
X.15     .  .
X.16     .  .
X.17     .  .
X.18     .  .
X.19     .  .
X.20     .  .

A = -2*E(11)-2*E(11)^2-2*E(11)^3-E(11)^4-E(11)^7-2*E(11)^8-2*E(11)^9-2*E(11)^10
B = -E(11)-2*E(11)^2-2*E(11)^3-2*E(11)^5-2*E(11)^6-2*E(11)^8-2*E(11)^9-E(11)^10
C = -2*E(11)^2-E(11)^3-2*E(11)^4-2*E(11)^5-2*E(11)^6-2*E(11)^7-E(11)^8-2*E(11)^9
D = -2*E(11)-2*E(11)^3-2*E(11)^4-E(11)^5-E(11)^6-2*E(11)^7-2*E(11)^8-2*E(11)^10
E = -2*E(11)-E(11)^2-2*E(11)^4-2*E(11)^5-2*E(11)^6-2*E(11)^7-E(11)^9-2*E(11)^10
F = E(11)^3+2*E(11)^4+E(11)^5+E(11)^6+2*E(11)^7+E(11)^8
G = 2*E(11)^2+E(11)^3+E(11)^4+E(11)^7+E(11)^8+2*E(11)^9
H = E(11)+2*E(11)^3+E(11)^5+E(11)^6+2*E(11)^8+E(11)^10
I = E(11)+E(11)^2+2*E(11)^5+2*E(11)^6+E(11)^9+E(11)^10
J = 2*E(11)+E(11)^2+E(11)^4+E(11)^7+E(11)^9+2*E(11)^10