Subgroup ($H$) information
| Description: | $C_7$ |
| Order: | \(7\) |
| Index: | \(131712\)\(\medspace = 2^{7} \cdot 3 \cdot 7^{3} \) |
| Exponent: | \(7\) |
| Generators: |
$\langle(1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,20,18,16,21,19,17)(22,24,26,28,23,25,27)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $D_7\wr S_4$ |
| Order: | \(921984\)\(\medspace = 2^{7} \cdot 3 \cdot 7^{4} \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^4.C_2^3.(C_6\times S_4)$, of order \(2765952\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7^{4} \) |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $32$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $D_7\wr S_4$ |