Subgroup ($H$) information
| Description: | $C_2^4:C_{12}$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(2\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
1 & 12 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 12 \\
12 & 1
\end{array}\right), \left(\begin{array}{rr}
13 & 0 \\
0 & 13
\end{array}\right), \left(\begin{array}{rr}
16 & 3 \\
15 & 16
\end{array}\right), \left(\begin{array}{rr}
19 & 0 \\
0 & 19
\end{array}\right), \left(\begin{array}{rr}
5 & 0 \\
0 & 23
\end{array}\right), \left(\begin{array}{rr}
1 & 16 \\
0 & 1
\end{array}\right)$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is normal, maximal, a semidirect factor, nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_2^6:C_6$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $1$ |
| 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
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_2^{12}.\PSL(2,7)$ |
| $\operatorname{Aut}(H)$ | $C_2^{10}.S_4$, of order \(24576\)\(\medspace = 2^{13} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2^{10}.S_4$, of order \(24576\)\(\medspace = 2^{13} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_2^4$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
| Centralizer: | $C_2^2\times C_6$ | ||
| Normalizer: | $C_2^6:C_6$ | ||
| Complements: | $C_2$ $C_2$ | ||
| Minimal over-subgroups: | $C_2^6:C_6$ | ||
| Maximal under-subgroups: | $C_2^3:C_{12}$ | $C_2^4\times C_6$ | $C_2^4:C_4$ |
Other information
| Number of subgroups in this autjugacy class | $7$ |
| Number of conjugacy classes in this autjugacy class | $7$ |
| Möbius function | $-1$ |
| Projective image | $C_2^4$ |