Subgroup ($H$) information
| Description: | $C_2^3\times C_8$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\left(\begin{array}{rr}
27 & 24 \\
8 & 19
\end{array}\right), \left(\begin{array}{rr}
1 & 16 \\
16 & 1
\end{array}\right), \left(\begin{array}{rr}
31 & 0 \\
0 & 31
\end{array}\right), \left(\begin{array}{rr}
17 & 0 \\
16 & 17
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $(C_4\times C_8).\GL(2,\mathbb{Z}/4)$ |
| Order: | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_6:C_8$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Automorphism Group: | $D_4\times D_6$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Outer Automorphisms: | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_4\times A_4).C_2^5.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_2.C_2^7:\GL(3,2)$, of order \(43008\)\(\medspace = 2^{11} \cdot 3 \cdot 7 \) |
| $\card{W}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
Other information
| Möbius function | not computed |
| Projective image | not computed |