Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(128\)\(\medspace = 2^{7} \) |
| Exponent: | \(3\) |
| Generators: |
$\left(\begin{array}{rr}
1 & 8 \\
0 & 1
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $3$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_6.C_2^6$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2^5\times C_4$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2^6.C_2^5.\GL(5,2)$, of order \(20478689280\)\(\medspace = 2^{21} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \) |
| Outer Automorphisms: | $C_2^6.C_2^5.\GL(5,2)$, of order \(20478689280\)\(\medspace = 2^{21} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6.C_2^6.C_2^5.\GL(5,2)$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(61436067840\)\(\medspace = 2^{21} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 31 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_2^5\times C_6$ | |
| Normalizer: | $C_6.C_2^6$ | |
| Minimal over-subgroups: | $C_6$ | $C_6$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_6.C_2^6$ |