Subgroup ($H$) information
| Description: | $C_3^5$ |
| Order: | \(243\)\(\medspace = 3^{5} \) |
| Index: | \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(7,8,9)(19,21,20)(31,32,33), (10,11,12)(22,24,23)(34,36,35), (4,5,6)(10,12,11) \!\cdots\! \rangle$
|
| 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). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_3^7.S_3^4:C_6$ |
| Order: | \(17006112\)\(\medspace = 2^{5} \cdot 3^{12} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_3^5:D_6\wr C_2$ |
| Order: | \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_3^2.C_3^5.C_2^3.C_2^4$, of order \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $4$ |
The quotient is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_3^4.C_6^3.C_2^3$, of order \(102036672\)\(\medspace = 2^{6} \cdot 3^{13} \) |
| $\operatorname{Aut}(H)$ | $\GL(5,3)$, of order \(475566474240\)\(\medspace = 2^{10} \cdot 3^{10} \cdot 5 \cdot 11^{2} \cdot 13 \) |
| $\card{W}$ | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_3^8$ |
| Normalizer: | $C_3^7.S_3^4:C_6$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^7.S_3^4:C_6$ |