Subgroup ($H$) information
| Description: | $Q_{64}\times C_{239}$ |
| Order: | \(15296\)\(\medspace = 2^{6} \cdot 239 \) |
| Index: | $1$ |
| Exponent: | \(7648\)\(\medspace = 2^{5} \cdot 239 \) |
| Generators: |
$b^{7170}, b^{956}, b^{5736}, b^{32}, b^{239}, a, b^{3824}$
|
| Nilpotency class: | $5$ |
| Derived length: | $2$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the radical, a direct factor, nonabelian, a Hall subgroup, elementary for $p = 2$ (hence hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $Q_{64}\times C_{239}$ |
| Order: | \(15296\)\(\medspace = 2^{6} \cdot 239 \) |
| Exponent: | \(7648\)\(\medspace = 2^{5} \cdot 239 \) |
| Nilpotency class: | $5$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $0$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{238}\times C_8.(C_8\times D_4)$ |
| $\operatorname{Aut}(H)$ | $C_{238}\times C_8.(C_8\times D_4)$ |
| $W$ | $D_{16}$, of order \(32\)\(\medspace = 2^{5} \) |
Related subgroups
| Centralizer: | $C_{478}$ | |||
| Normalizer: | $Q_{64}\times C_{239}$ | |||
| Complements: | $C_1$ | |||
| Maximal under-subgroups: | $Q_{32}\times C_{239}$ | $Q_{32}\times C_{239}$ | $C_{7648}$ | $Q_{64}$ |
Other information
| Möbius function | $1$ |
| Projective image | $D_{16}$ |