Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(224\)\(\medspace = 2^{5} \cdot 7 \) |
| Exponent: | \(3\) |
| Generators: |
$d^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $3$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $D_{42}:C_2^3$ |
| Order: | \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^2\times D_{28}$ |
| Order: | \(224\)\(\medspace = 2^{5} \cdot 7 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Automorphism Group: | $F_7\times C_2^6:(C_2\times S_4)$, of order \(129024\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 7 \) |
| Outer Automorphisms: | $C_2^5.(C_6\times S_4)$ |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4.C_2^4.C_7.C_3^3.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(193536\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 7 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{42}:C_2^3$ | ||||
| Normalizer: | $D_{42}:C_2^3$ | ||||
| Complements: | $C_2^2\times D_{28}$ | ||||
| Minimal over-subgroups: | $C_{21}$ | $C_6$ | $C_6$ | $C_6$ | $S_3$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $D_{42}:C_2^3$ |