Subgroup ($H$) information
| Description: | $D_7\wr A_4$ |
| Order: | \(460992\)\(\medspace = 2^{6} \cdot 3 \cdot 7^{4} \) |
| Index: | \(35\)\(\medspace = 5 \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(29,30,31,34,35,32,33), (16,19)(17,18)(20,21)(23,26)(24,25)(27,28), (1,15,5,20,4,19,7,17,6,18,3,16,2,21) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_7^5:(C_2^4:A_5)$ |
| Order: | \(16134720\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 7^{5} \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and perfect (hence nonsolvable).
Quotient set structure
Since this subgroup has trivial core, the ambient group $G$ acts faithfully and transitively on the set of cosets of $H$. The resulting permutation representation is isomorphic to 35T164.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^5:(C_2\wr S_5\times C_3)$, of order \(193616640\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \cdot 7^{5} \) |
| $\operatorname{Aut}(H)$ | $C_7^4.C_2^3.(C_6\times S_4)$, of order \(2765952\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7^{4} \) |
| $W$ | $D_7\wr A_4$, of order \(460992\)\(\medspace = 2^{6} \cdot 3 \cdot 7^{4} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $D_7\wr A_4$ |
| Normal closure: | $C_7^5:(C_2^4:A_5)$ |
| Core: | $C_1$ |
Other information
| Number of subgroups in this autjugacy class | $35$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_7^5:(C_2^4:A_5)$ |