Properties

Label 44T12
Order \(176\)
n \(44\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_4\times D_{11}$

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Group action invariants

Degree $n$ :  $44$
Transitive number $t$ :  $12$
Group :  $D_4\times D_{11}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,27)(26,28)(29,31)(30,32)(33,36)(34,35)(37,39)(38,40)(41,44)(42,43), (1,36)(2,35)(3,34)(4,33)(5,30)(6,29)(7,32)(8,31)(9,26)(10,25)(11,28)(12,27)(13,23)(14,24)(15,21)(16,22)(17,18)(19,20)(37,43)(38,44)(39,42)(40,41), (1,6,2,5)(3,8,4,7)(9,43,10,44)(11,41,12,42)(13,39,14,40)(15,37,16,38)(17,36,18,35)(19,33,20,34)(21,31,22,32)(23,29,24,30)(25,28,26,27)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$
16:  $D_4\times C_2$
22:  $D_{11}$
44:  $D_{22}$ x 3

Resolvents shown for degrees $\leq 29$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 11: $D_{11}$

Degree 22: $D_{22}$

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

Group invariants

Order:  $176=2^{4} \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [176, 31]
Character table: Data not available.