# Properties

 Label 25T16 Degree $25$ Order $150$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_5^2:S_3$

# Learn more about

## Group action invariants

 Degree $n$: $25$ Transitive number $t$: $16$ Group: $C_5^2:S_3$ Parity: $1$ Primitive: yes Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,12,8)(2,6,13)(3,5,18)(4,24,23)(9,25,17)(10,19,22)(11,14,21)(15,20,16), (1,2,3,4,5)(6,21,8,23,10,25,7,22,9,24)(11,20,13,17,15,19,12,16,14,18)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$

Resolvents shown for degrees $\leq 47$

Degree 5: None

## Low degree siblings

15T13, 15T14, 30T37, 30T38

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $15$ $2$ $( 6,25)( 7,21)( 8,22)( 9,23)(10,24)(11,19)(12,20)(13,16)(14,17)(15,18)$ $3, 3, 3, 3, 3, 3, 3, 3, 1$ $50$ $3$ $( 2, 6,25)( 3,11,19)( 4,16,13)( 5,21, 7)( 8,10,20)( 9,15,14)(12,24,22) (17,18,23)$ $5, 5, 5, 5, 5$ $3$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ $10, 10, 5$ $15$ $10$ $( 1, 2, 3, 4, 5)( 6,21, 8,23,10,25, 7,22, 9,24)(11,20,13,17,15,19,12,16,14,18)$ $10, 10, 5$ $15$ $10$ $( 1, 2, 7, 8,13,14,19,20,25,21)( 3,12, 9,18,15,24,16, 5,22, 6)( 4,17,10,23,11)$ $5, 5, 5, 5, 5$ $3$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$ $10, 10, 5$ $15$ $10$ $( 1, 3,13,15,25,22, 7, 9,19,16)( 2, 8,14,20,21)( 4,18,11, 5,23,12,10,24,17, 6)$ $5, 5, 5, 5, 5$ $3$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)$ $10, 10, 5$ $15$ $10$ $( 1, 4, 2, 5, 3)( 6,23, 7,24, 8,25, 9,21,10,22)(11,17,12,18,13,19,14,20,15,16)$ $5, 5, 5, 5, 5$ $3$ $5$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)$ $5, 5, 5, 5, 5$ $6$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$ $5, 5, 5, 5, 5$ $6$ $5$ $( 1, 9,12,20,23)( 2,10,13,16,24)( 3, 6,14,17,25)( 4, 7,15,18,21) ( 5, 8,11,19,22)$

## Group invariants

 Order: $150=2 \cdot 3 \cdot 5^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [150, 5]
 Character table:  2 1 1 . 1 1 1 1 1 1 1 1 . . 3 1 . 1 . . . . . . . . . . 5 2 1 . 2 1 1 2 1 2 1 2 2 2 1a 2a 3a 5a 10a 10b 5b 10c 5c 10d 5d 5e 5f 2P 1a 1a 3a 5b 5b 5d 5d 5c 5a 5a 5c 5f 5e 3P 1a 2a 1a 5c 10d 10a 5a 10b 5d 10c 5b 5f 5e 5P 1a 2a 3a 1a 2a 2a 1a 2a 1a 2a 1a 1a 1a 7P 1a 2a 3a 5b 10b 10c 5d 10d 5a 10a 5c 5f 5e X.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 X.3 2 . -1 2 . . 2 . 2 . 2 2 2 X.4 3 -1 . A D /E /B /D B E /A F *F X.5 3 -1 . B E D A /E /A /D /B *F F X.6 3 -1 . /A /D E B D /B /E A F *F X.7 3 -1 . /B /E /D /A E A D B *F F X.8 3 1 . A -D -/E /B -/D B -E /A F *F X.9 3 1 . B -E -D A -/E /A -/D /B *F F X.10 3 1 . /A -/D -E B -D /B -/E A F *F X.11 3 1 . /B -/E -/D /A -E A -D B *F F X.12 6 . . C . . *C . *C . C G *G X.13 6 . . *C . . C . C . *C *G G A = 2*E(5)^3+E(5)^4 B = E(5)^2+2*E(5)^4 C = -2*E(5)-2*E(5)^4 = 1-Sqrt(5) = 1-r5 D = -E(5)^4 E = -E(5)^2 F = -E(5)^2-E(5)^3 = (1+Sqrt(5))/2 = 1+b5 G = E(5)+2*E(5)^2+2*E(5)^3+E(5)^4 = (-3-Sqrt(5))/2 = -2-b5