Subgroup ($H$) information
| Description: | $(C_3^2\times \He_3):C_4$ |
| Order: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$b^{6}c^{2}dfg, f, e, b^{4}d^{2}ef^{2}, g, cdf^{2}g^{2}, de^{2}fg^{2}$
|
| Derived length: | $3$ |
The subgroup is the commutator subgroup (hence characteristic and normal), nonabelian, and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $(C_3^2\times \He_3):\SD_{16}$ |
| Order: | \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_3.C_4:S_3^2.C_2$, of order \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $C_3:S_3.C_6^2.C_{12}.C_2^3$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_3^4:(C_2\times \SD_{16})$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3\) |
| $W$ | $C_3^4:\SD_{16}$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
Related subgroups
Other information
| Möbius function | $2$ |
| Projective image | $(C_3^2\times \He_3):\SD_{16}$ |