Subgroup ($H$) information
| Description: | $C_3^3:C_{12}$ |
| Order: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$c^{18}e^{2}, e, de, b^{2}c^{8}de, c^{8}de, c^{12}de^{2}$
|
| Derived length: | $2$ |
The subgroup is the commutator subgroup (hence characteristic and normal), nonabelian, monomial (hence solvable), metabelian, and an A-group.
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: | $C_2^2\times C_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2^3:S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Outer Automorphisms: | $C_2^3:S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
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 \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $\PSU(3,2):C_2\times \GL(2,3)$, of order \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
| $W$ | $C_4\times F_9:C_2$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_3^4:(C_4\times \SD_{16})$ |