Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Exponent: | \(2\) |
| Generators: |
$b, c^{7}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $D_7:A_4$ |
| Order: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $F_7$ |
| Order: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Automorphism Group: | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4\times F_7$, of order \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_3$, of order \(3\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $W$ | $C_3$, of order \(3\) |
Related subgroups
| Centralizer: | $C_2\times D_{14}$ | ||
| Normalizer: | $D_7:A_4$ | ||
| Complements: | $F_7$ | ||
| Minimal over-subgroups: | $C_2\times C_{14}$ | $A_4$ | $C_2^3$ |
| Maximal under-subgroups: | $C_2$ |
Other information
| Möbius function | $-7$ |
| Projective image | $D_7:A_4$ |