# Properties

 Label 30T4 Order $$30$$ n $$30$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_3\times D_5$

## Group action invariants

 Degree $n$ : $30$ Transitive number $t$ : $4$ Group : $C_3\times D_5$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,27,11,8,21,18)(2,28,12,7,22,17)(3,6,13,16,24,25)(4,5,14,15,23,26)(9,30,19,10,29,20), (1,19,7,26,13)(2,20,8,25,14)(3,21,9,28,15)(4,22,10,27,16)(5,24,11,29,17)(6,23,12,30,18) $|\Aut(F/K)|$: $30$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
10:  $D_{5}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 5: $D_{5}$

Degree 6: $C_6$

Degree 10: $D_5$

Degree 15: $D_5\times C_3$

## Low degree siblings

15T3

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

## Group invariants

 Order: $30=2 \cdot 3 \cdot 5$ Cyclic: No Abelian: No Solvable: Yes GAP id: [30, 2]
 Character table:  2 1 1 . 1 . 1 . 1 . . 1 . 3 1 1 1 1 1 1 1 1 1 1 1 1 5 1 . 1 . 1 . 1 1 1 1 1 1 1a 2a 15a 6a 15b 6b 5a 3a 5b 15c 3b 15d 2P 1a 1a 15b 3a 15a 3b 5b 3b 5a 15d 3a 15c 3P 1a 2a 5a 2a 5b 2a 5b 1a 5a 5b 1a 5a 5P 1a 2a 3a 6b 3b 6a 1a 3b 1a 3a 3a 3b 7P 1a 2a 15c 6a 15d 6b 5b 3a 5a 15a 3b 15b 11P 1a 2a 15d 6b 15c 6a 5a 3b 5b 15b 3a 15a 13P 1a 2a 15c 6a 15d 6b 5b 3a 5a 15a 3b 15b X.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 X.3 1 -1 A -A /A -/A 1 /A 1 A A /A X.4 1 -1 /A -/A A -A 1 A 1 /A /A A X.5 1 1 A A /A /A 1 /A 1 A A /A X.6 1 1 /A /A A A 1 A 1 /A /A A X.7 2 . B . *B . *B 2 B *B 2 B X.8 2 . *B . B . B 2 *B B 2 *B X.9 2 . C . /D . *B E B D /E /C X.10 2 . /C . D . *B /E B /D E C X.11 2 . D . /C . B E *B C /E /D X.12 2 . /D . C . B /E *B /C E D A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 C = E(15)^11+E(15)^14 D = E(15)^2+E(15)^8 E = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3