# Properties

 Label 30T2 Degree $30$ Order $30$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_5\times S_3$

# Related objects

## Group action invariants

 Degree $n$: $30$ Transitive number $t$: $2$ Group: $C_5\times S_3$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $30$ Generators: (1,22,12)(2,21,11)(3,24,14)(4,23,13)(5,26,16)(6,25,15)(7,28,18)(8,27,17)(9,30,19)(10,29,20), (1,27,14,9,26,21,7,4,20,15)(2,28,13,10,25,22,8,3,19,16)(5,11,18,23,29,6,12,17,24,30)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$6$:  $S_3$
$10$:  $C_{10}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: $C_5$

Degree 6: $S_3$

Degree 10: $C_{10}$

Degree 15: $S_3 \times C_5$

## Low degree siblings

15T4

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

## Group invariants

 Order: $30=2 \cdot 3 \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [30, 1]
 Character table:  2 1 1 . 1 . 1 1 1 1 . . 1 . 1 1 3 1 . 1 . 1 . 1 . . 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1a 2a 15a 10a 15b 10b 5a 10c 10d 15c 3a 5b 15d 5c 5d 2P 1a 1a 15b 5d 15c 5c 5b 5b 5a 15d 3a 5d 15a 5a 5c 3P 1a 2a 5a 10c 5b 10a 5c 10d 10b 5d 1a 5a 5c 5d 5b 5P 1a 2a 3a 2a 3a 2a 1a 2a 2a 3a 3a 1a 3a 1a 1a 7P 1a 2a 15b 10b 15c 10d 5b 10a 10c 15d 3a 5d 15a 5a 5c 11P 1a 2a 15a 10a 15b 10b 5a 10c 10d 15c 3a 5b 15d 5c 5d 13P 1a 2a 15d 10c 15a 10a 5c 10d 10b 15b 3a 5a 15c 5d 5b X.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 X.3 1 -1 A -A B -B /B -/B -/A /A 1 A /B /A B X.4 1 -1 B -B /A -/A A -A -/B /B 1 B A /B /A X.5 1 -1 /B -/B A -A /A -/A -B B 1 /B /A B A X.6 1 -1 /A -/A /B -/B B -B -A A 1 /A B A /B X.7 1 1 A A B B /B /B /A /A 1 A /B /A B X.8 1 1 B B /A /A A A /B /B 1 B A /B /A X.9 1 1 /B /B A A /A /A B B 1 /B /A B A X.10 1 1 /A /A /B /B B B A A 1 /A B A /B X.11 2 . -1 . -1 . 2 . . -1 -1 2 -1 2 2 X.12 2 . -/B . -A . C . . -B -1 /D -/A D /C X.13 2 . -/A . -/B . D . . -A -1 C -B /C /D X.14 2 . -A . -B . /D . . -/A -1 /C -/B C D X.15 2 . -B . -/A . /C . . -/B -1 D -A /D C A = E(5)^4 B = E(5)^3 C = 2*E(5) D = 2*E(5)^3