Subgroup ($H$) information
| Description: | $C_3:C_{116}$ |
| Order: | \(348\)\(\medspace = 2^{2} \cdot 3 \cdot 29 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(348\)\(\medspace = 2^{2} \cdot 3 \cdot 29 \) |
| Generators: |
$abc^{145}, c^{6}, c^{116}, b^{2}$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $D_6\times C_{116}$ |
| Order: | \(1392\)\(\medspace = 2^{4} \cdot 3 \cdot 29 \) |
| Exponent: | \(348\)\(\medspace = 2^{2} \cdot 3 \cdot 29 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{21}:(C_2^4.C_2^4)$ |
| $\operatorname{Aut}(H)$ | $D_6\times C_{28}$, of order \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(S)$ | $D_6\times C_{28}$, of order \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
Other information
| Möbius function | $2$ |
| Projective image | $C_2\times D_6$ |