Subgroup ($H$) information
| Description: | $C_2^2\times Q_8$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 2 & 1 & 1 \\
1 & 2 & 1 & 2 \\
0 & 0 & 1 & 0 \\
2 & 0 & 1 & 1
\end{array}\right), \left(\begin{array}{rrrr}
1 & 1 & 2 & 2 \\
2 & 1 & 1 & 0 \\
0 & 0 & 1 & 0 \\
1 & 2 & 0 & 2
\end{array}\right), \left(\begin{array}{rrrr}
2 & 0 & 2 & 0 \\
0 & 2 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 1 & 2
\end{array}\right), \left(\begin{array}{rrrr}
2 & 0 & 0 & 0 \\
0 & 2 & 0 & 0 \\
0 & 0 & 2 & 0 \\
0 & 0 & 0 & 2
\end{array}\right), \left(\begin{array}{rrrr}
2 & 0 & 2 & 0 \\
1 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
1 & 1 & 1 & 0
\end{array}\right)$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), a semidirect factor, nonabelian, a $p$-group (hence elementary and hyperelementary), metabelian, and rational.
Ambient group ($G$) information
| Description: | $C_2^2\times \GL(2,3)$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^3:S_4^2$, of order \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \) |
| $\operatorname{Aut}(H)$ | $C_2^6:(S_3\times S_4)$, of order \(9216\)\(\medspace = 2^{10} \cdot 3^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_4^2$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $3$ |
| Projective image | $S_4$ |