Subgroup ($H$) information
| Description: | $C_7\wr A_4$ |
| Order: | \(28812\)\(\medspace = 2^{2} \cdot 3 \cdot 7^{4} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$b^{2}c^{7}d^{7}e^{5}f^{3}, e, c^{7}d^{4}, d^{7}ef, c^{2}f^{6}, f, d^{2}$
|
| Derived length: | $3$ |
The subgroup is the commutator subgroup (hence characteristic and normal), a semidirect factor, nonabelian, solvable, and an A-group. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $D_7\times C_7^3:S_4$ |
| Order: | \(115248\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{4} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^3.(C_7\times A_4).C_6^2.C_2$ |
| $\operatorname{Aut}(H)$ | $C_4\times S_3\times S_5$, of order \(296352\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 7^{3} \) |
| $W$ | $C_2\times C_7^3:S_4$, of order \(16464\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{3} \) |
Related subgroups
Other information
| Möbius function | $2$ |
| Projective image | $D_7\times C_7^3:S_4$ |