Subgroup ($H$) information
| Description: | $D_7\wr S_4$ |
| Order: | \(921984\)\(\medspace = 2^{7} \cdot 3 \cdot 7^{4} \) |
| Index: | $1$ |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(1,28,18,13,2,26,16,8)(3,24,21,10,7,23,20,11)(4,22,19,12,6,25,15,9)(5,27,17,14) \!\cdots\! \rangle$
|
| Derived length: | $5$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, and a Hall subgroup. Whether it is monomial has not been computed.
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.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^4.C_2^3.(C_6\times S_4)$, of order \(2765952\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7^{4} \) |
| $\operatorname{Aut}(H)$ | $C_7^4.C_2^3.(C_6\times S_4)$, of order \(2765952\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7^{4} \) |
| $W$ | $D_7\wr S_4$, of order \(921984\)\(\medspace = 2^{7} \cdot 3 \cdot 7^{4} \) |
Related subgroups
| Centralizer: | $C_1$ | |||||
| Normalizer: | $D_7\wr S_4$ | |||||
| Complements: | $C_1$ | |||||
| Maximal under-subgroups: | $D_7\wr A_4$ | $C_7:D_7^3:S_4$ | $C_7^4:Q_8:S_4$ | $D_7\wr D_4$ | $C_7^3.A_4.C_2^2\times D_7$ | $C_2\wr S_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $D_7\wr S_4$ |