Subgroup ($H$) information
| Description: | $C_{15}:Q_8$ |
| Order: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$bc^{3}, c^{12}, c^{8}, c^{6}, d^{2}$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{60}.C_2^4$ |
| 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: | $C_2^3$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{15}:(C_2^3.C_2^6.C_2)$ |
| $\operatorname{Aut}(H)$ | $S_3\times D_4\times F_5$, of order \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| $\operatorname{res}(S)$ | $S_3\times D_4\times F_5$, of order \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $D_6:D_{10}$, of order \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Möbius function | $-8$ |
| Projective image | $D_{30}:C_2^3$ |