Subgroup ($H$) information
| Description: | $C_{12}:Q_8$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a, c^{8}, c^{12}, b, c^{18}, b^{2}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_{20}.(S_3\times Q_8)$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 \(2\) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{15}:(C_2^2.C_2^6.C_2^3)$, of order \(30720\)\(\medspace = 2^{11} \cdot 3 \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2^6:D_6$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{12}:C_2^5$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| $W$ | $S_3\times D_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Related subgroups
Other information
| Möbius function | $5$ |
| Projective image | $D_6:D_{10}$ |