# Properties

 Label 25T8 Order $$100$$ n $$25$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $D_{25}.C_2$

## Group action invariants

 Degree $n$ : $25$ Transitive number $t$ : $8$ Group : $D_{25}.C_2$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,12,7,23)(2,14,6,21)(3,11,10,24)(4,13,9,22)(5,15,8,25)(16,20,18,19), (1,14)(2,13)(3,12)(4,11)(5,15)(6,8)(9,10)(16,23)(17,22)(18,21)(19,25)(20,24) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
20:  $F_5$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 5: $F_5$

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

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

## Group invariants

 Order: $100=2^{2} \cdot 5^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [100, 3]
 Character table:  2 2 2 2 2 . . . . . . 5 2 . . . 2 2 2 2 2 2 1a 4a 4b 2a 5a 25a 25b 25c 25d 25e 2P 1a 2a 2a 1a 5a 25e 25a 25b 25c 25d 3P 1a 4b 4a 2a 5a 25d 25e 25a 25b 25c 5P 1a 4a 4b 2a 1a 5a 5a 5a 5a 5a 7P 1a 4b 4a 2a 5a 25a 25b 25c 25d 25e 11P 1a 4b 4a 2a 5a 25e 25a 25b 25c 25d 13P 1a 4a 4b 2a 5a 25b 25c 25d 25e 25a 17P 1a 4a 4b 2a 5a 25c 25d 25e 25a 25b 19P 1a 4b 4a 2a 5a 25c 25d 25e 25a 25b 23P 1a 4b 4a 2a 5a 25e 25a 25b 25c 25d X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 1 1 1 1 1 X.3 1 A -A -1 1 1 1 1 1 1 X.4 1 -A A -1 1 1 1 1 1 1 X.5 4 . . . 4 -1 -1 -1 -1 -1 X.6 4 . . . -1 B D C F E X.7 4 . . . -1 C F E B D X.8 4 . . . -1 D C F E B X.9 4 . . . -1 E B D C F X.10 4 . . . -1 F E B D C A = -E(4) = -Sqrt(-1) = -i B = -E(25)^4-E(25)^6+E(25)^7-E(25)^9-E(25)^11-E(25)^14-E(25)^16+E(25)^18-E(25)^19-E(25)^21 C = E(25)^6+E(25)^8+E(25)^17+E(25)^19 D = E(25)^9+E(25)^12+E(25)^13+E(25)^16 E = -E(25)^3-E(25)^7-E(25)^8+E(25)^11-E(25)^12-E(25)^13+E(25)^14-E(25)^17-E(25)^18-E(25)^22 F = E(25)^3+E(25)^4+E(25)^21+E(25)^22