Subgroup ($H$) information
| Description: | $C_2^3\times C_{10}$ |
| Order: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$a, c^{5}, d^{5}, b^{2}, c^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $C_{10}^2.C_2^3$ |
| Order: | \(800\)\(\medspace = 2^{5} \cdot 5^{2} \) |
| 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: | $D_5$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^5\times D_5).C_2^5.\PSL(2,7)$ |
| $\operatorname{Aut}(H)$ | $C_4\times A_8$, of order \(80640\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_4\times C_2^3:\GL(3,2)$, of order \(5376\)\(\medspace = 2^{8} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{10}^2.C_2^3$ | ||
| Normalizer: | $C_{10}^2.C_2^3$ | ||
| Minimal over-subgroups: | $C_2^2\times C_{10}^2$ | $C_2^3\times C_{20}$ | |
| Maximal under-subgroups: | $C_2^2\times C_{10}$ | $C_2^2\times C_{10}$ | $C_2^4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $5$ |
| Projective image | $D_5$ |