Properties

Label 36T22
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_6^2:C_2$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $22$
Group :  $C_6^2:C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,2,5)(3,8,4,7)(9,24,10,23)(11,22,12,21)(13,30,14,29)(15,32,16,31)(17,26,18,25)(19,27,20,28)(33,36,34,35), (1,28,16,2,27,15)(3,26,13)(4,25,14)(5,32,19)(6,31,20)(7,30,17,8,29,18)(9,34,22,10,33,21)(11,35,23)(12,36,24), (1,10,29)(2,9,30)(3,12,31,4,11,32)(5,13,24,6,14,23)(7,16,21)(8,15,22)(17,27,34)(18,28,33)(19,26,36,20,25,35)
$|\Aut(F/K)|$:  $18$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 4
8:  $D_{4}$
12:  $D_{6}$ x 4
18:  $C_3^2:C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 4

Degree 4: $D_{4}$

Degree 6: $S_3$ x 4

Degree 9: $C_3^2:C_2$

Degree 12: $(C_6\times C_2):C_2$ x 4

Degree 18: $C_3^2 : C_2$

Low degree siblings

There are no siblings with degree $\leq 10$
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 $ $1$ $1$ $()$
$ 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 $ $2$ $2$ $( 3, 4)( 5, 6)(11,12)(13,14)(19,20)(23,24)(25,26)(31,32)(35,36)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $18$ $2$ $( 1, 3)( 2, 4)( 5,33)( 6,34)( 7,35)( 8,36)( 9,32)(10,31)(11,29)(12,30)(13,27) (14,28)(15,25)(16,26)(17,23)(18,24)(19,22)(20,21)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $18$ $4$ $( 1, 3, 2, 4)( 5,34, 6,33)( 7,35, 8,36)( 9,32,10,31)(11,30,12,29)(13,28,14,27) (15,25,16,26)(17,23,18,24)(19,21,20,22)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $ $2$ $6$ $( 1, 7,34)( 2, 8,33)( 3, 5,35, 4, 6,36)( 9,15,18)(10,16,17)(11,14,20,12,13,19) (21,27,29)(22,28,30)(23,25,31,24,26,32)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 7,34)( 2, 8,33)( 3, 6,35)( 4, 5,36)( 9,15,18)(10,16,17)(11,13,20) (12,14,19)(21,27,29)(22,28,30)(23,26,31)(24,25,32)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1, 8,34, 2, 7,33)( 3, 5,35, 4, 6,36)( 9,16,18,10,15,17)(11,14,20,12,13,19) (21,28,29,22,27,30)(23,25,31,24,26,32)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $ $2$ $6$ $( 1, 8,34, 2, 7,33)( 3, 6,35)( 4, 5,36)( 9,16,18,10,15,17)(11,13,20)(12,14,19) (21,28,29,22,27,30)(23,26,31)(24,25,32)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $ $2$ $6$ $( 1, 9,29, 2,10,30)( 3,11,31)( 4,12,32)( 5,14,24)( 6,13,23)( 7,15,21, 8,16,22) (17,28,34,18,27,33)(19,25,36)(20,26,35)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1, 9,29, 2,10,30)( 3,12,31, 4,11,32)( 5,13,24, 6,14,23)( 7,15,21, 8,16,22) (17,28,34,18,27,33)(19,26,36,20,25,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,10,29)( 2, 9,30)( 3,11,31)( 4,12,32)( 5,14,24)( 6,13,23)( 7,16,21) ( 8,15,22)(17,27,34)(18,28,33)(19,25,36)(20,26,35)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $ $2$ $6$ $( 1,10,29)( 2, 9,30)( 3,12,31, 4,11,32)( 5,13,24, 6,14,23)( 7,16,21)( 8,15,22) (17,27,34)(18,28,33)(19,26,36,20,25,35)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $ $2$ $6$ $( 1,15,27, 2,16,28)( 3,13,26)( 4,14,25)( 5,19,32)( 6,20,31)( 7,18,29, 8,17,30) ( 9,21,33,10,22,34)(11,23,35)(12,24,36)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,15,27, 2,16,28)( 3,14,26, 4,13,25)( 5,20,32, 6,19,31)( 7,18,29, 8,17,30) ( 9,21,33,10,22,34)(11,24,35,12,23,36)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,16,27)( 2,15,28)( 3,13,26)( 4,14,25)( 5,19,32)( 6,20,31)( 7,17,29) ( 8,18,30)( 9,22,33)(10,21,34)(11,23,35)(12,24,36)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $ $2$ $6$ $( 1,16,27)( 2,15,28)( 3,14,26, 4,13,25)( 5,20,32, 6,19,31)( 7,17,29)( 8,18,30) ( 9,22,33)(10,21,34)(11,24,35,12,23,36)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $ $2$ $6$ $( 1,17,21)( 2,18,22)( 3,19,23, 4,20,24)( 5,11,25, 6,12,26)( 7,10,27)( 8, 9,28) (13,32,35,14,31,36)(15,30,33)(16,29,34)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1,17,21)( 2,18,22)( 3,20,23)( 4,19,24)( 5,12,25)( 6,11,26)( 7,10,27) ( 8, 9,28)(13,31,35)(14,32,36)(15,30,33)(16,29,34)$
$ 6, 6, 6, 6, 6, 6 $ $2$ $6$ $( 1,18,21, 2,17,22)( 3,19,23, 4,20,24)( 5,11,25, 6,12,26)( 7, 9,27, 8,10,28) (13,32,35,14,31,36)(15,29,33,16,30,34)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $ $2$ $6$ $( 1,18,21, 2,17,22)( 3,20,23)( 4,19,24)( 5,12,25)( 6,11,26)( 7, 9,27, 8,10,28) (13,31,35)(14,32,36)(15,29,33,16,30,34)$

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 35]
Character table: Data not available.