# Properties

 Label 38T6 Degree $38$ Order $228$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2\times D_{19}:C_3$

## Group action invariants

 Degree $n$: $38$ Transitive number $t$: $6$ Group: $C_2\times D_{19}:C_3$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $2$ Generators: (1,11,6,2,12,5)(3,25,28,4,26,27)(7,15,33,8,16,34)(9,29,18,10,30,17)(13,20,24,14,19,23)(21,38,35,22,37,36)(31,32), (1,3)(2,4)(5,37)(6,38)(7,35)(8,36)(9,33)(10,34)(11,31)(12,32)(13,29)(14,30)(15,28)(16,27)(17,25)(18,26)(19,23)(20,24)(21,22)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$12$:  $C_6\times C_2$
$114$:  $C_{19}:C_{6}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 19: $C_{19}:C_{6}$

## Low degree siblings

38T6

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

## Group invariants

 Order: $228=2^{2} \cdot 3 \cdot 19$ Cyclic: no Abelian: no Solvable: yes GAP id: [228, 7]
 Character table:  2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 . . . . . . 19 1 . . . . . 1 . . . . . 1 1 1 1 1 1 1a 3a 6a 3b 6b 2a 2b 6c 6d 6e 6f 2c 38a 19a 38b 19b 19c 38c 2P 1a 3b 3a 3a 3b 1a 1a 3b 3a 3a 3b 1a 19b 19b 19c 19c 19a 19a 3P 1a 1a 2a 1a 2a 2a 2b 2b 2c 2b 2c 2c 38b 19b 38c 19c 19a 38a 5P 1a 3b 6b 3a 6a 2a 2b 6e 6f 6c 6d 2c 38b 19b 38c 19c 19a 38a 7P 1a 3a 6a 3b 6b 2a 2b 6c 6d 6e 6f 2c 38a 19a 38b 19b 19c 38c 11P 1a 3b 6b 3a 6a 2a 2b 6e 6f 6c 6d 2c 38a 19a 38b 19b 19c 38c 13P 1a 3a 6a 3b 6b 2a 2b 6c 6d 6e 6f 2c 38c 19c 38a 19a 19b 38b 17P 1a 3b 6b 3a 6a 2a 2b 6e 6f 6c 6d 2c 38b 19b 38c 19c 19a 38a 19P 1a 3a 6a 3b 6b 2a 2b 6c 6d 6e 6f 2c 2b 1a 2b 1a 1a 2b 23P 1a 3b 6b 3a 6a 2a 2b 6e 6f 6c 6d 2c 38c 19c 38a 19a 19b 38b 29P 1a 3b 6b 3a 6a 2a 2b 6e 6f 6c 6d 2c 38c 19c 38a 19a 19b 38b 31P 1a 3a 6a 3b 6b 2a 2b 6c 6d 6e 6f 2c 38a 19a 38b 19b 19c 38c 37P 1a 3a 6a 3b 6b 2a 2b 6c 6d 6e 6f 2c 38a 19a 38b 19b 19c 38c X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 1 -1 -1 -1 -1 1 -1 1 1 -1 1 -1 1 1 -1 X.3 1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 1 1 1 1 1 X.4 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 -1 1 1 -1 X.5 1 A -/A /A -A -1 -1 -A /A -/A A 1 -1 1 -1 1 1 -1 X.6 1 /A -A A -/A -1 -1 -/A A -A /A 1 -1 1 -1 1 1 -1 X.7 1 A -/A /A -A -1 1 A -/A /A -A -1 1 1 1 1 1 1 X.8 1 /A -A A -/A -1 1 /A -A A -/A -1 1 1 1 1 1 1 X.9 1 A /A /A A 1 -1 -A -/A -/A -A -1 -1 1 -1 1 1 -1 X.10 1 /A A A /A 1 -1 -/A -A -A -/A -1 -1 1 -1 1 1 -1 X.11 1 A /A /A A 1 1 A /A /A A 1 1 1 1 1 1 1 X.12 1 /A A A /A 1 1 /A A A /A 1 1 1 1 1 1 1 X.13 6 . . . . . 6 . . . . . B B C C D D X.14 6 . . . . . 6 . . . . . C C D D B B X.15 6 . . . . . 6 . . . . . D D B B C C X.16 6 . . . . . -6 . . . . . -B B -C C D -D X.17 6 . . . . . -6 . . . . . -C C -D D B -B X.18 6 . . . . . -6 . . . . . -D D -B B C -C A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(19)^2+E(19)^3+E(19)^5+E(19)^14+E(19)^16+E(19)^17 C = E(19)^4+E(19)^6+E(19)^9+E(19)^10+E(19)^13+E(19)^15 D = E(19)+E(19)^7+E(19)^8+E(19)^11+E(19)^12+E(19)^18