Subgroup ($H$) information
| Description: | $C_3:\OD_{64}$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | $1$ |
| Exponent: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Generators: |
$a, b^{24}, b^{12}, b^{3}, b^{16}, a^{2}, b^{30}$
|
| Derived length: | $2$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, metacyclic (hence supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_3:\OD_{64}$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{12}.C_2^5$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_{12}.C_2^5$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $W$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_{16}$ | |||
| Normalizer: | $C_3:\OD_{64}$ | |||
| Complements: | $C_1$ | |||
| Maximal under-subgroups: | $C_2\times C_{48}$ | $C_3:C_{32}$ | $C_3:C_{32}$ | $\OD_{64}$ |
Other information
| Möbius function | $1$ |
| Projective image | $D_6$ |