Subgroup ($H$) information
| Description: | $C_5^3:C_2$ |
| Order: | \(250\)\(\medspace = 2 \cdot 5^{3} \) |
| Index: | \(3\) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$a^{3}, c, d, b$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, a Hall subgroup, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_5^3:C_6$ |
| Order: | \(750\)\(\medspace = 2 \cdot 3 \cdot 5^{3} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_5\times F_{25}:C_2$, of order \(24000\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $\AGL(3,5)$, of order \(186000000\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{6} \cdot 31 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $F_5\times F_{25}:C_2$, of order \(24000\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $C_5^3:C_6$, of order \(750\)\(\medspace = 2 \cdot 3 \cdot 5^{3} \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_5^3:C_6$ |