Subgroup ($H$) information
| Description: | $Q_{16}\times C_{33}$ |
| Order: | \(528\)\(\medspace = 2^{4} \cdot 3 \cdot 11 \) |
| Index: | \(2\) |
| Exponent: | \(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \) |
| Generators: |
$a, b^{8}, c^{22}, c^{3}, b^{14}, b^{4}$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), maximal, nonabelian, elementary for $p = 2$ (hence hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $C_{33}:Q_{32}$ |
| Order: | \(1056\)\(\medspace = 2^{5} \cdot 3 \cdot 11 \) |
| Exponent: | \(528\)\(\medspace = 2^{4} \cdot 3 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{44}.(C_2^4\times C_{20})$ |
| $\operatorname{Aut}(H)$ | $C_5:(C_4.C_2^5)$, of order \(640\)\(\medspace = 2^{7} \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_5 \times (C_8:C_2^4)$, of order \(640\)\(\medspace = 2^{7} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(22\)\(\medspace = 2 \cdot 11 \) |
| $W$ | $D_8$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
| Centralizer: | $C_{66}$ | |||
| Normalizer: | $C_{33}:Q_{32}$ | |||
| Minimal over-subgroups: | $C_{33}:Q_{32}$ | |||
| Maximal under-subgroups: | $C_{264}$ | $Q_8\times C_{33}$ | $C_{11}\times Q_{16}$ | $C_3\times Q_{16}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_{11}:D_8$ |