Subgroup ($H$) information
| Description: | $C_5:D_5$ |
| Order: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rrr}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 2 & 1
\end{array}\right), \left(\begin{array}{rrr}
1 & 4 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right), \left(\begin{array}{rrr}
1 & 0 & 0 \\
0 & 4 & 0 \\
0 & 2 & 1
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_2\times C_5^2:Q_8$ |
| Order: | \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), and rational.
Quotient group ($Q$) structure
| Description: | $Q_8$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{10}^2:\Unitary(2,3)$, of order \(9600\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $C_5^2.\GL(2,5)$, of order \(12000\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{3} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_5^2:\Unitary(2,3)$, of order \(2400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_5^2:Q_8$, of order \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
Related subgroups
| Centralizer: | $C_2$ | |||
| Normalizer: | $C_2\times C_5^2:Q_8$ | |||
| Complements: | $Q_8$ $Q_8$ $Q_8$ $Q_8$ | |||
| Minimal over-subgroups: | $C_5:D_{10}$ | |||
| Maximal under-subgroups: | $C_5^2$ | $D_5$ | $D_5$ | $D_5$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_2\times C_5^2:Q_8$ |