Subgroup ($H$) information
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(39600\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\left(\begin{array}{rrrr}
4 & 6 & 9 & 0 \\
8 & 7 & 0 & 2 \\
4 & 0 & 7 & 6 \\
0 & 7 & 8 & 4
\end{array}\right), \left(\begin{array}{rrrr}
6 & 9 & 9 & 1 \\
2 & 1 & 10 & 7 \\
7 & 1 & 2 & 7 \\
8 & 0 & 1 & 0
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a $p$-group.
Ambient group ($G$) information
| Description: | $C_{15}:Q_8\times \SL(2,11)$ |
| Order: | \(158400\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_5\times S_3\times \SL(2,11)$ |
| Order: | \(39600\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Automorphism Group: | $C_2\times C_4\times \PSL(2,11).C_2\times S_3$ |
| Outer Automorphisms: | $C_2^2\times C_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^5\times S_3).C_2^2.\PSL(2,11).C_2$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Möbius function | not computed |
| Projective image | not computed |