Subgroup ($H$) information
| Description: | $C_5:C_4$ |
| Order: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Index: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$c^{5}, d, c^{10}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{10}.D_5^2$ |
| Order: | \(1000\)\(\medspace = 2^{3} \cdot 5^{3} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_5:D_5$ |
| Order: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_5^2.\GL(2,5)$, of order \(12000\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{3} \) |
| Outer Automorphisms: | $A_5:C_4$, of order \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(D_5\times C_5:D_5).C_2^4.S_5$ |
| $\operatorname{Aut}(H)$ | $C_2\times F_5$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times F_5$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(24000\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{3} \) |
| $W$ | $D_5$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
Related subgroups
| Centralizer: | $C_5^2:C_4$ | |
| Normalizer: | $C_{10}.D_5^2$ | |
| Complements: | $C_5:D_5$ $C_5:D_5$ | |
| Minimal over-subgroups: | $C_5:C_{20}$ | $C_4\times D_5$ |
| Maximal under-subgroups: | $C_{10}$ | $C_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-125$ |
| Projective image | $C_5:D_5^2$ |