Subgroup ($H$) information
| Description: | $C_3^2\times \He_3$ |
| Order: | \(243\)\(\medspace = 3^{5} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(3\) |
| Generators: |
$b, c, de, ef$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is the commutator subgroup (hence characteristic and normal), the Fitting subgroup (hence nilpotent, solvable, supersolvable, and monomial), a semidirect factor, nonabelian, a $3$-Sylow subgroup (hence a Hall subgroup), a $p$-group (hence elementary and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $(C_3^2\times \He_3):C_4$ |
| Order: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:S_3.C_6^2.C_{12}.C_2^3$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $C_3^6.(C_3^2:\GL(2,3)\times \GL(2,3))$, of order \(15116544\)\(\medspace = 2^{8} \cdot 3^{10} \) |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(9\)\(\medspace = 3^{2} \) |
| $W$ | $C_3^2:C_4$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_3^3$ | ||
| Normalizer: | $(C_3^2\times \He_3):C_4$ | ||
| Complements: | $C_4$ | ||
| Minimal over-subgroups: | $C_3^4:S_3$ | ||
| Maximal under-subgroups: | $C_3\times \He_3$ | $C_3\times \He_3$ | $C_3^4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_3^4:C_4$ |