# Properties

 Label 25T31 Order $$400$$ n $$25$$ Cyclic No Abelian No Solvable Yes Primitive Yes $p$-group No Group: $D_5^2.C_4$

## Group action invariants

 Degree $n$ : $25$ Transitive number $t$ : $31$ Group : $D_5^2.C_4$ Parity: $-1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,12,19,5,7,18,25,6,13,24)(2,17,20,10,8,23,21,11,14,4)(3,22,16,15,9), (1,21,18,8,22,2,10,20)(3,19,17,24,25,9,6,4)(5,12,16,15,23,11,7,13) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $C_4\times C_2$
16:  $C_8:C_2$

Resolvents shown for degrees $\leq 47$

Degree 5: None

## Low degree siblings

10T28, 20T104, 20T107, 20T109, 20T115, 40T397, 40T398, 40T399, 40T400

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 4, 4, 4, 4, 4, 1$ $25$ $4$ $( 2, 3, 5, 4)( 6,11,21,16)( 7,13,25,19)( 8,15,24,17)( 9,12,23,20)(10,14,22,18)$ $4, 4, 4, 4, 4, 4, 1$ $25$ $4$ $( 2, 4, 5, 3)( 6,16,21,11)( 7,19,25,13)( 8,17,24,15)( 9,20,23,12)(10,18,22,14)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1$ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $10$ $2$ $( 2, 6)( 3,11)( 4,16)( 5,21)( 8,12)( 9,17)(10,22)(14,18)(15,23)(20,24)$ $8, 8, 8, 1$ $50$ $8$ $( 2, 8, 4,17, 5,24, 3,15)( 6,20,16,23,21,12,11, 9)( 7,22,19,14,25,10,13,18)$ $8, 8, 8, 1$ $50$ $8$ $( 2, 9, 4,20, 5,23, 3,12)( 6,15,16, 8,21,17,11,24)( 7,18,19,22,25,14,13,10)$ $4, 4, 4, 4, 4, 4, 1$ $50$ $4$ $( 2,11, 5,16)( 3,21, 4, 6)( 7,13,25,19)( 8,23,24, 9)(10,18,22,14)(12,15,20,17)$ $8, 8, 8, 1$ $50$ $8$ $( 2,12, 3,23, 5,20, 4, 9)( 6,24,11,17,21, 8,16,15)( 7,10,13,14,25,22,19,18)$ $8, 8, 8, 1$ $50$ $8$ $( 2,15, 3,24, 5,17, 4, 8)( 6, 9,11,12,21,23,16,20)( 7,18,13,10,25,14,19,22)$ $5, 5, 5, 5, 5$ $16$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ $10, 10, 5$ $40$ $10$ $( 1, 2, 7, 8,13,14,19,20,25,21)( 3,12, 9,18,15,24,16, 5,22, 6)( 4,17,10,23,11)$ $5, 5, 5, 5, 5$ $8$ $5$ $( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)$

## Group invariants

 Order: $400=2^{4} \cdot 5^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [400, 206]
 Character table:  2 4 4 4 4 3 3 3 3 3 3 . 1 1 5 2 . . . 1 . . . . . 2 1 2 1a 4a 4b 2a 2b 8a 8b 4c 8c 8d 5a 10a 5b 2P 1a 2a 2a 1a 1a 4b 4b 2a 4a 4a 5a 5b 5b 3P 1a 4b 4a 2a 2b 8d 8c 4c 8b 8a 5a 10a 5b 5P 1a 4a 4b 2a 2b 8a 8b 4c 8c 8d 1a 2b 1a 7P 1a 4b 4a 2a 2b 8d 8c 4c 8b 8a 5a 10a 5b 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 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 X.5 1 -1 -1 1 -1 B B 1 -B -B 1 -1 1 X.6 1 -1 -1 1 -1 -B -B 1 B B 1 -1 1 X.7 1 -1 -1 1 1 B -B -1 B -B 1 1 1 X.8 1 -1 -1 1 1 -B B -1 -B B 1 1 1 X.9 2 A -A -2 . . . . . . 2 . 2 X.10 2 -A A -2 . . . . . . 2 . 2 X.11 8 . . . -4 . . . . . -2 1 3 X.12 8 . . . 4 . . . . . -2 -1 3 X.13 16 . . . . . . . . . 1 . -4 A = -2*E(4) = -2*Sqrt(-1) = -2i B = -E(4) = -Sqrt(-1) = -i