Group action invariants
| Degree $n$ : | $25$ | |
| Transitive number $t$ : | $12$ | |
| Group : | $D_5^2$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,24,16,14,6,4,21,19,11,9)(2,23,17,13,7,3,22,18,12,8)(5,25,20,15,10), (1,7)(2,6)(3,10)(4,9)(5,8)(11,22)(12,21)(13,25)(14,24)(15,23)(16,17)(18,20) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 10: $D_{5}$ x 2 20: $D_{10}$ x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $D_{5}$ x 2
Low degree siblings
10T9 x 2, 20T28 x 2Siblings 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 $ | $5$ | $2$ | $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)(15,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $25$ | $2$ | $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$ |
| $ 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$ |
| $ 10, 10, 5 $ | $10$ | $10$ | $( 1, 2, 3, 4, 5)( 6,22, 8,24,10,21, 7,23, 9,25)(11,17,13,19,15,16,12,18,14,20)$ |
| $ 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$ |
| $ 10, 10, 5 $ | $10$ | $10$ | $( 1, 3, 5, 2, 4)( 6,23,10,22, 9,21, 8,25, 7,24)(11,18,15,17,14,16,13,20,12,19)$ |
| $ 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 6,11,16,21)( 2, 7,12,17,22)( 3, 8,13,18,23)( 4, 9,14,19,24) ( 5,10,15,20,25)$ |
| $ 10, 10, 5 $ | $10$ | $10$ | $( 1, 6,11,16,21)( 2,10,12,20,22, 5, 7,15,17,25)( 3, 9,13,19,23, 4, 8,14,18,24)$ |
| $ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)$ |
| $ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$ |
| $ 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,11,21, 6,16)( 2,12,22, 7,17)( 3,13,23, 8,18)( 4,14,24, 9,19) ( 5,15,25,10,20)$ |
| $ 10, 10, 5 $ | $10$ | $10$ | $( 1,11,21, 6,16)( 2,15,22,10,17, 5,12,25, 7,20)( 3,14,23, 9,18, 4,13,24, 8,19)$ |
| $ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1,12,23, 9,20)( 2,13,24,10,16)( 3,14,25, 6,17)( 4,15,21, 7,18) ( 5,11,22, 8,19)$ |
| $ 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1,13,25, 7,19)( 2,14,21, 8,20)( 3,15,22, 9,16)( 4,11,23,10,17) ( 5,12,24, 6,18)$ |
Group invariants
| Order: | $100=2^{2} \cdot 5^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [100, 13] |
| Character table: |
2 2 2 2 2 1 1 1 1 1 1 . . 1 1 . .
5 2 1 1 . 2 1 2 1 2 1 2 2 2 1 2 2
1a 2a 2b 2c 5a 10a 5b 10b 5c 10c 5d 5e 5f 10d 5g 5h
2P 1a 1a 1a 1a 5b 5b 5a 5a 5f 5f 5h 5g 5c 5c 5e 5d
3P 1a 2a 2b 2c 5b 10b 5a 10a 5f 10d 5h 5g 5c 10c 5e 5d
5P 1a 2a 2b 2c 1a 2a 1a 2a 1a 2b 1a 1a 1a 2b 1a 1a
7P 1a 2a 2b 2c 5b 10b 5a 10a 5f 10d 5h 5g 5c 10c 5e 5d
X.1 1 1 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 -1 1 1
X.3 1 -1 1 -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 -1 1 1
X.5 2 -2 . . A -A *A -*A 2 . A *A 2 . A *A
X.6 2 -2 . . *A -*A A -A 2 . *A A 2 . *A A
X.7 2 . -2 . 2 . 2 . A -A A A *A -*A *A *A
X.8 2 . -2 . 2 . 2 . *A -*A *A *A A -A A A
X.9 2 . 2 . 2 . 2 . A A A A *A *A *A *A
X.10 2 . 2 . 2 . 2 . *A *A *A *A A A A A
X.11 2 2 . . A A *A *A 2 . A *A 2 . A *A
X.12 2 2 . . *A *A A A 2 . *A A 2 . *A A
X.13 4 . . . B . *B . B . C -1 *B . -1 *C
X.14 4 . . . *B . B . *B . *C -1 B . -1 C
X.15 4 . . . B . *B . *B . -1 *C B . C -1
X.16 4 . . . *B . B . B . -1 C *B . *C -1
A = E(5)^2+E(5)^3
= (-1-Sqrt(5))/2 = -1-b5
B = 2*E(5)^2+2*E(5)^3
= -1-Sqrt(5) = -1-r5
C = -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4
= (3+Sqrt(5))/2 = 2+b5
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