Subgroup ($H$) information
| Description: | $A_4:Q_8$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Index: | \(2\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
12 & 3 \\
3 & 16
\end{array}\right), \left(\begin{array}{rr}
15 & 0 \\
14 & 15
\end{array}\right), \left(\begin{array}{rr}
1 & 14 \\
14 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 12 \\
8 & 13
\end{array}\right), \left(\begin{array}{rr}
13 & 0 \\
0 & 13
\end{array}\right), \left(\begin{array}{rr}
15 & 7 \\
21 & 8
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $A_4:Q_{16}$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\GL(2,\mathbb{Z}/4):C_2^2$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $D_4\times S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_4\times S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $\GL(2,\mathbb{Z}/4)$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2$ | |||
| Normalizer: | $A_4:Q_{16}$ | |||
| Minimal over-subgroups: | $A_4:Q_{16}$ | |||
| Maximal under-subgroups: | $C_4\times A_4$ | $A_4:C_4$ | $C_2^2:Q_8$ | $C_3:Q_8$ |
Other information
| Möbius function | $-1$ |
| Projective image | $\GL(2,\mathbb{Z}/4)$ |