Subgroup ($H$) information
| Description: | $C_{17}:C_4$ |
| Order: | \(68\)\(\medspace = 2^{2} \cdot 17 \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(68\)\(\medspace = 2^{2} \cdot 17 \) |
| Generators: |
$ac, d^{102}, d^{12}$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{204}.C_2^3$ |
| Order: | \(1632\)\(\medspace = 2^{5} \cdot 3 \cdot 17 \) |
| Exponent: | \(204\)\(\medspace = 2^{2} \cdot 3 \cdot 17 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^2\times C_6$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_2\times \GL(3,2)$, of order \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_2\times \GL(3,2)$, of order \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4.C_2^4.C_{51}.C_8.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2\times F_{17}$, of order \(544\)\(\medspace = 2^{5} \cdot 17 \) |
| $\operatorname{res}(S)$ | $C_2\times F_{17}$, of order \(544\)\(\medspace = 2^{5} \cdot 17 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $W$ | $D_{34}$, of order \(68\)\(\medspace = 2^{2} \cdot 17 \) |
Related subgroups
| Centralizer: | $C_2^2\times C_6$ | ||
| Normalizer: | $C_{204}.C_2^3$ | ||
| Minimal over-subgroups: | $C_{17}:C_{12}$ | $C_{34}:C_4$ | $C_{17}:Q_8$ |
| Maximal under-subgroups: | $C_{34}$ | $C_4$ |
Other information
| Number of subgroups in this autjugacy class | $8$ |
| Number of conjugacy classes in this autjugacy class | $8$ |
| Möbius function | $8$ |
| Projective image | $C_{102}:C_2^3$ |