Subgroup ($H$) information
| Description: | $C_2^3\times D_6$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
43 & 30 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
25 & 4 \\
16 & 29
\end{array}\right), \left(\begin{array}{rr}
43 & 22 \\
22 & 43
\end{array}\right), \left(\begin{array}{rr}
1 & 22 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
21 & 22 \\
22 & 21
\end{array}\right), \left(\begin{array}{rr}
1 & 22 \\
22 & 1
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, an A-group, and rational.
Ambient group ($G$) information
| Description: | $C_2^2:D_{12}\times C_{10}$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_{10}$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_2^6.C_2^5.C_2^4)$ |
| $\operatorname{Aut}(H)$ | $C_2^4.A_8\times S_3$, of order \(1935360\)\(\medspace = 2^{11} \cdot 3^{3} \cdot 5 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^5.(D_4\times D_6)$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(32\)\(\medspace = 2^{5} \) |
| $W$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_{10}\times D_6$ |