Subgroup ($H$) information
| Description: | $C_6:C_8$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$a^{4}, c, b, b^{2}, b^{4}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $C_4^2.D_6$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{12}:C_2^6$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $D_4\times D_6$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2\times D_6$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_4^2$ | |||
| Normalizer: | $C_4^2.D_6$ | |||
| Minimal over-subgroups: | $C_{12}:C_8$ | |||
| Maximal under-subgroups: | $C_2\times C_{12}$ | $C_3:C_8$ | $C_3:C_8$ | $C_2\times C_8$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_6.D_4$ |