Group action invariants
| Degree $n$ : | $25$ | |
| Transitive number $t$ : | $9$ | |
| Group : | $C_5^2:C_4$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,14,24,6,17)(2,15,25,7,18)(3,11,21,8,19)(4,12,22,9,20)(5,13,23,10,16), (1,25,19,12,10)(2,21,20,13,6)(3,22,16,14,7)(4,23,17,15,8)(5,24,18,11,9), (1,19,25,10)(2,16,24,8)(3,18,23,6)(4,20,22,9)(5,17,21,7)(11,15,13,14) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 4: $C_4$ 20: $F_5$ x 6 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $F_5$ x 6
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 $ | $25$ | $4$ | $( 2, 3, 5, 4)( 6,14,24,17)( 7,11,23,20)( 8,13,22,18)( 9,15,21,16)(10,12,25,19)$ |
| $ 4, 4, 4, 4, 4, 4, 1 $ | $25$ | $4$ | $( 2, 4, 5, 3)( 6,17,24,14)( 7,20,23,11)( 8,18,22,13)( 9,16,21,15)(10,19,25,12)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $25$ | $2$ | $( 2, 5)( 3, 4)( 6,24)( 7,23)( 8,22)( 9,21)(10,25)(11,20)(12,19)(13,18)(14,17) (15,16)$ |
| $ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ |
| $ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 6,14,17,24)( 2, 7,15,18,25)( 3, 8,11,19,21)( 4, 9,12,20,22) ( 5,10,13,16,23)$ |
| $ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 7,11,20,23)( 2, 8,12,16,24)( 3, 9,13,17,25)( 4,10,14,18,21) ( 5, 6,15,19,22)$ |
| $ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 8,13,18,22)( 2, 9,14,19,23)( 3,10,15,20,24)( 4, 6,11,16,25) ( 5, 7,12,17,21)$ |
| $ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 9,15,16,21)( 2,10,11,17,22)( 3, 6,12,18,23)( 4, 7,13,19,24) ( 5, 8,14,20,25)$ |
| $ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1,10,12,19,25)( 2, 6,13,20,21)( 3, 7,14,16,22)( 4, 8,15,17,23) ( 5, 9,11,18,24)$ |
Group invariants
| Order: | $100=2^{2} \cdot 5^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [100, 11] |
| Character table: |
2 2 2 2 2 . . . . . .
5 2 . . . 2 2 2 2 2 2
1a 4a 4b 2a 5a 5b 5c 5d 5e 5f
2P 1a 2a 2a 1a 5a 5b 5c 5d 5e 5f
3P 1a 4b 4a 2a 5a 5b 5c 5d 5e 5f
5P 1a 4a 4b 2a 1a 1a 1a 1a 1a 1a
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 4 -1 -1 -1 -1
X.7 4 . . . -1 -1 4 -1 -1 -1
X.8 4 . . . -1 -1 -1 4 -1 -1
X.9 4 . . . -1 -1 -1 -1 -1 4
X.10 4 . . . -1 -1 -1 -1 4 -1
A = -E(4)
= -Sqrt(-1) = -i
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