Group action invariants
| Degree $n$: | $10$ | |
| Transitive number $t$: | $42$ | |
| Group: | $A_5^2 : C_4$ | |
| CHM label: | $1/2[S(5)^{2}]2$ | |
| Parity: | $1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $\card{\Aut(F/K)}$: | $1$ | |
| Generators: | (2,4,6,8,10), (1,6)(2,5,10,7)(3,8)(4,9) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 5: None
Low degree siblings
12T278, 20T457, 20T461, 24T12116, 25T100, 30T817, 36T9861, 40T10509, 40T10510, 40T10511Siblings 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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1 $ | $100$ | $2$ | $(1,3)(6,8)$ |
| $ 2, 2, 2, 2, 1, 1 $ | $225$ | $2$ | $( 1, 3)( 2,10)( 5, 7)( 6, 8)$ |
| $ 3, 3, 1, 1, 1, 1 $ | $400$ | $3$ | $(1,3,5)(2,6,8)$ |
| $ 3, 3, 2, 2 $ | $400$ | $6$ | $( 1, 3, 5)( 2, 6, 8)( 4,10)( 7, 9)$ |
| $ 4, 4, 1, 1 $ | $900$ | $4$ | $( 1, 3, 5, 7)( 2,10, 6, 8)$ |
| $ 5, 5 $ | $576$ | $5$ | $( 1, 3, 5, 7, 9)( 2,10, 4, 6, 8)$ |
| $ 2, 2, 1, 1, 1, 1, 1, 1 $ | $30$ | $2$ | $( 2,10)( 6, 8)$ |
| $ 3, 1, 1, 1, 1, 1, 1, 1 $ | $40$ | $3$ | $(2,6,8)$ |
| $ 5, 1, 1, 1, 1, 1 $ | $48$ | $5$ | $( 2,10, 4, 6, 8)$ |
| $ 3, 2, 2, 1, 1, 1 $ | $400$ | $6$ | $( 1, 3)( 2, 6, 8)( 4,10)$ |
| $ 4, 2, 1, 1, 1, 1 $ | $600$ | $4$ | $( 1, 3)( 2,10, 6, 8)$ |
| $ 3, 2, 2, 1, 1, 1 $ | $600$ | $6$ | $(1,3)(2,6,8)(5,7)$ |
| $ 5, 2, 2, 1 $ | $720$ | $10$ | $( 1, 3)( 2,10, 4, 6, 8)( 5, 7)$ |
| $ 5, 3, 1, 1 $ | $960$ | $15$ | $( 1, 3, 5)( 2,10, 4, 6, 8)$ |
| $ 4, 3, 2, 1 $ | $1200$ | $12$ | $( 1, 3, 5)( 2,10, 6, 8)( 7, 9)$ |
| $ 4, 2, 2, 2 $ | $600$ | $4$ | $( 1, 6)( 2, 5)( 3, 8)( 4, 7,10, 9)$ |
| $ 6, 4 $ | $1200$ | $12$ | $( 1, 8, 3, 2, 5, 6)( 4, 7,10, 9)$ |
| $ 8, 2 $ | $1800$ | $8$ | $( 1, 4, 7, 8, 3,10, 9, 6)( 2, 5)$ |
| $ 4, 2, 2, 2 $ | $600$ | $4$ | $( 1, 6)( 2, 5)( 3, 8)( 4, 9,10, 7)$ |
| $ 6, 4 $ | $1200$ | $12$ | $( 1, 8, 3, 2, 5, 6)( 4, 9,10, 7)$ |
| $ 8, 2 $ | $1800$ | $8$ | $( 1,10, 7, 8, 3, 4, 9, 6)( 2, 5)$ |
Group invariants
| Order: | $14400=2^{6} \cdot 3^{2} \cdot 5^{2}$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | no | |
| Label: | not available |
| Character table: not available. |