Subgroup ($H$) information
| Description: | $C_7:C_3$ |
| Order: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Generators: |
$c^{4}, d^{2}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a Hall subgroup, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 3$.
Ambient group ($G$) information
| Description: | $C_2^4.F_7$ |
| 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), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2^3\times C_4$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2^4:C_2^3:\GL(3,2)$, of order \(21504\)\(\medspace = 2^{10} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_2^4:C_2^3:\GL(3,2)$, of order \(21504\)\(\medspace = 2^{10} \cdot 3 \cdot 7 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_7\times C_2^4:C_2^3:\GL(3,2)$, of order \(903168\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 7^{2} \) |
| $\operatorname{Aut}(H)$ | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(21504\)\(\medspace = 2^{10} \cdot 3 \cdot 7 \) |
| $W$ | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_2^4$ | |
| Normalizer: | $C_2^4.F_7$ | |
| Complements: | $C_2^3\times C_4$ | |
| Minimal over-subgroups: | $C_7:C_6$ | $C_7:C_6$ |
| Maximal under-subgroups: | $C_7$ | $C_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_2^4.F_7$ |