Properties

Label 36T18
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 $t$ :  $18$
Group :  $C_6\times A_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,30,10,2,29,9)(3,32,12,4,31,11)(5,23,15,6,24,16)(7,22,14,8,21,13)(17,36,27)(18,35,28)(19,33,25)(20,34,26), (1,21,20,4,23,18)(2,22,19,3,24,17)(5,27,9,8,25,12)(6,28,10,7,26,11)(13,33,31,15,36,30)(14,34,32,16,35,29)
$|\Aut(F/K)|$:  $12$

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: $C_2$

Degree 3: $C_3$ x 4

Degree 4: None

Degree 6: $C_6$ x 4, $A_4$, $A_4\times C_2$

Degree 9: $C_3^2$

Degree 12: $A_4 \times C_2$

Degree 18: $C_6 \times C_3$, $A_4 \times C_3$, 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 $ $3$ $2$ $( 5, 6)( 7, 8)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(33,34) (35,36)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,24) (22,23)(25,27)(26,28)(29,31)(30,32)(33,36)(34,35)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,23) (22,24)(25,27)(26,28)(29,32)(30,31)(33,36)(34,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 5,33)( 2, 6,34)( 3, 7,35)( 4, 8,36)( 9,16,20)(10,15,19)(11,13,17) (12,14,18)(21,28,31)(22,27,32)(23,26,30)(24,25,29)$
$ 6, 6, 6, 6, 6, 6 $ $4$ $6$ $( 1, 7,33, 4, 6,36)( 2, 8,34, 3, 5,35)( 9,13,20,12,15,18)(10,14,19,11,16,17) (21,25,32,23,27,29)(22,26,31,24,28,30)$
$ 6, 6, 6, 6, 3, 3, 3, 3 $ $3$ $6$ $( 1, 9,29, 2,10,30)( 3,11,31, 4,12,32)( 5,15,24)( 6,16,23)( 7,14,21)( 8,13,22) (17,28,36,18,27,35)(19,26,33,20,25,34)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,10,29)( 2, 9,30)( 3,12,31)( 4,11,32)( 5,15,24)( 6,16,23)( 7,14,21) ( 8,13,22)(17,27,36)(18,28,35)(19,25,33)(20,26,34)$
$ 6, 6, 6, 6, 6, 6 $ $1$ $6$ $( 1,11,29, 4,10,32)( 2,12,30, 3, 9,31)( 5,13,24, 8,15,22)( 6,14,23, 7,16,21) (17,25,36,19,27,33)(18,26,35,20,28,34)$
$ 6, 6, 6, 6, 6, 6 $ $3$ $6$ $( 1,11,29, 4,10,32)( 2,12,30, 3, 9,31)( 5,14,24, 7,15,21)( 6,13,23, 8,16,22) (17,26,36,20,27,34)(18,25,35,19,28,33)$
$ 6, 6, 6, 6, 6, 6 $ $4$ $6$ $( 1,13,25, 4,15,27)( 2,14,26, 3,16,28)( 5,17,29, 8,19,32)( 6,18,30, 7,20,31) ( 9,21,34,12,23,35)(10,22,33,11,24,36)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,15,25)( 2,16,26)( 3,14,28)( 4,13,27)( 5,19,29)( 6,20,30)( 7,18,31) ( 8,17,32)( 9,23,34)(10,24,33)(11,22,36)(12,21,35)$
$ 6, 6, 6, 6, 6, 6 $ $4$ $6$ $( 1,17,24, 4,19,22)( 2,18,23, 3,20,21)( 5,11,25, 8,10,27)( 6,12,26, 7, 9,28) (13,29,36,15,32,33)(14,30,35,16,31,34)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,19,23)( 2,20,24)( 3,18,22)( 4,17,21)( 5, 9,26)( 6,10,25)( 7,11,27) ( 8,12,28)(13,31,35)(14,32,36)(15,30,34)(16,29,33)$
$ 6, 6, 6, 6, 6, 6 $ $4$ $6$ $( 1,21,20, 4,23,18)( 2,22,19, 3,24,17)( 5,27, 9, 8,25,12)( 6,28,10, 7,26,11) (13,33,31,15,36,30)(14,34,32,16,35,29)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,23,20)( 2,24,19)( 3,22,17)( 4,21,18)( 5,25, 9)( 6,26,10)( 7,28,11) ( 8,27,12)(13,36,31)(14,35,32)(15,33,30)(16,34,29)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,25,15)( 2,26,16)( 3,28,14)( 4,27,13)( 5,29,19)( 6,30,20)( 7,31,18) ( 8,32,17)( 9,34,23)(10,33,24)(11,36,22)(12,35,21)$
$ 6, 6, 6, 6, 6, 6 $ $4$ $6$ $( 1,27,16, 4,25,14)( 2,28,15, 3,26,13)( 5,31,20, 8,30,18)( 6,32,19, 7,29,17) ( 9,35,24,12,34,22)(10,36,23,11,33,21)$
$ 6, 6, 6, 6, 3, 3, 3, 3 $ $3$ $6$ $( 1,29,10)( 2,30, 9)( 3,31,12)( 4,32,11)( 5,23,15, 6,24,16)( 7,22,14, 8,21,13) (17,35,27,18,36,28)(19,34,25,20,33,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,29,10)( 2,30, 9)( 3,31,12)( 4,32,11)( 5,24,15)( 6,23,16)( 7,21,14) ( 8,22,13)(17,36,27)(18,35,28)(19,33,25)(20,34,26)$
$ 6, 6, 6, 6, 6, 6 $ $3$ $6$ $( 1,31,10, 3,29,12)( 2,32, 9, 4,30,11)( 5,21,15, 7,24,14)( 6,22,16, 8,23,13) (17,33,27,19,36,25)(18,34,28,20,35,26)$
$ 6, 6, 6, 6, 6, 6 $ $1$ $6$ $( 1,32,10, 4,29,11)( 2,31, 9, 3,30,12)( 5,22,15, 8,24,13)( 6,21,16, 7,23,14) (17,33,27,19,36,25)(18,34,28,20,35,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1,33, 5)( 2,34, 6)( 3,35, 7)( 4,36, 8)( 9,20,16)(10,19,15)(11,17,13) (12,18,14)(21,31,28)(22,32,27)(23,30,26)(24,29,25)$
$ 6, 6, 6, 6, 6, 6 $ $4$ $6$ $( 1,35, 6, 4,34, 7)( 2,36, 5, 3,33, 8)( 9,17,15,12,19,13)(10,18,16,11,20,14) (21,29,28,23,32,26)(22,30,27,24,31,25)$

Group invariants

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