Properties

Label 36T31
Order \(72\)
n \(36\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_6\times A_4$

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Group action invariants

Degree $n$ :  $36$
Transitive number $t$ :  $31$
Group :  $C_6\times A_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,20,30,2,19,29)(3,18,32,4,17,31)(5,22,33,6,21,34)(7,24,35,8,23,36)(9,15,27,10,16,28)(11,13,25,12,14,26), (1,15,36)(2,16,35)(3,14,33)(4,13,34)(5,20,26)(6,19,25)(7,17,28)(8,18,27)(9,22,32)(10,21,31)(11,24,29)(12,23,30)
$|\Aut(F/K)|$:  $12$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$ x 4
6:  $C_6$ x 4
9:  $C_3^2$
12:  $A_4$
24:  $A_4\times C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: None

Degree 3: $C_3$ x 4

Degree 4: None

Degree 6: $A_4$, $A_4\times C_2$

Degree 9: $C_3^2$

Degree 12: $A_4\times C_2$

Degree 18: $A_4 \times C_3$, 18T25

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 47]
Character table: Data not available.