Subgroup ($H$) information
| Description: | $C_3^4:(C_4\times \SD_{16})$ |
| Order: | \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| Index: | $1$ |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$a, de, c^{18}e^{2}, c^{12}e, e, b^{3}, a^{2}, c^{8}de, b^{2}c^{16}d^{2}e^{2}, c^{3}$
|
| Derived length: | $3$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, and monomial.
Ambient group ($G$) information
| Description: | $C_3^4:(C_4\times \SD_{16})$ |
| Order: | \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_8^2.C_2^2$, of order \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $C_3^4.C_8^2.C_2^2$, of order \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| $W$ | $C_3^4:(C_4\times \SD_{16})$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
Related subgroups
Other information
| Möbius function | $1$ |
| Projective image | $C_3^4:(C_4\times \SD_{16})$ |