# Properties

 Label 36T45 Degree $36$ Order $72$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_4\times D_9$

## Group action invariants

 Degree $n$: $36$ Transitive number $t$: $45$ Group: $C_4\times D_9$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $4$ Generators: (1,18,2,17)(3,19,4,20)(5,14,6,13)(7,16,8,15)(9,11,10,12)(21,33,22,34)(23,35,24,36)(25,29,26,30)(27,31,28,32), (1,23)(2,24)(3,22)(4,21)(5,20)(6,19)(7,17)(8,18)(9,16)(10,15)(11,13)(12,14)(25,33)(26,34)(27,35)(28,36)(29,30)(31,32)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$
$18$:  $D_{9}$
$24$:  $S_3 \times C_4$
$36$:  $D_{18}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$

Degree 6: $D_{6}$

Degree 9: $D_{9}$

Degree 12: $S_3 \times C_4$

Degree 18: $D_{18}$

## Low degree siblings

36T45

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

## Group invariants

 Order: $72=2^{3} \cdot 3^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [72, 5]
 Character table: not available.