Subgroup ($H$) information
| Description: | $C_3^6$ |
| Order: | \(729\)\(\medspace = 3^{6} \) |
| Index: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(5,13,6), (1,12,4)(5,6,13)(9,17,11), (5,13,6)(8,16,10), (2,7,3)(5,13,6)(14,15,18), (5,6,13)(14,18,15), (9,11,17)\rangle$
|
| 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: | $C_3^8.(C_4\times S_4)$ |
| Order: | \(629856\)\(\medspace = 2^{5} \cdot 3^{9} \) |
| 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: | $C_3^2:(C_4\times S_4)$ |
| Order: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2^2\times C_6^2:D_6$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian and monomial (hence solvable).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^7.A_4.C_{12}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $\GL(6,3)$, of order \(84129611558952960\)\(\medspace = 2^{13} \cdot 3^{15} \cdot 5 \cdot 7 \cdot 11^{2} \cdot 13^{2} \) |
| $W$ | $C_4\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_3^8$ |
| Normalizer: | $C_3^8.(C_4\times S_4)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^8.(C_4\times S_4)$ |