Subgroup ($H$) information
| Description: | $C_3^5$ |
| Order: | \(243\)\(\medspace = 3^{5} \) |
| Index: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Exponent: | \(3\) |
| Generators: |
$a^{2}, f^{3}, cd^{2}, d^{2}, e^{3}f^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $C_3^5.S_3^2:C_2^2$ |
| Order: | \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $S_3^2:C_2^2$ |
| Order: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_6^2:\SD_{16}$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian, monomial (hence solvable), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_9^2.(C_3\times C_6).C_4.C_6.C_2^3$, of order \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $\GL(5,3)$, of order \(475566474240\)\(\medspace = 2^{10} \cdot 3^{10} \cdot 5 \cdot 11^{2} \cdot 13 \) |
| $W$ | $S_3^2:C_2^2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_3^5$ | |||||
| Normalizer: | $C_3^5.S_3^2:C_2^2$ | |||||
| Minimal over-subgroups: | $C_9:C_3^4$ | $C_3^4:C_6$ | $C_3^4:C_6$ | $C_3^4:C_6$ | $C_3^4:C_6$ | |
| Maximal under-subgroups: | $C_3^4$ | $C_3^4$ | $C_3^4$ | $C_3^4$ | $C_3^4$ | $C_3^4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_3^5.S_3^2:C_2^2$ |