Subgroup ($H$) information
| Description: | $C_3\times D_{56}$ |
| Order: | \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Generators: |
$a, d^{2}, c^{8}, c^{18}, c^{3}d^{7}, c^{12}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $D_{56}.D_6$ |
| Order: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_{84}).C_6.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_8:C_2^3\times F_7$, of order \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \) |
| $\card{W}$ | \(112\)\(\medspace = 2^{4} \cdot 7 \) |
Related subgroups
| Centralizer: | $C_{12}$ | |||
| Normalizer: | $D_{56}.D_6$ | |||
| Minimal over-subgroups: | $D_{56}:C_6$ | $D_{56}:S_3$ | ||
| Maximal under-subgroups: | $C_3\times D_{28}$ | $C_{168}$ | $D_{56}$ | $C_3\times D_8$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |