Subgroup ($H$) information
| Description: | $C_3^7$ |
| Order: | \(2187\)\(\medspace = 3^{7} \) |
| Index: | \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(10,11,12)(22,24,23)(34,35,36), (1,2,3)(7,8,9)(13,15,14)(19,21,20)(25,27,26) \!\cdots\! \rangle$
|
| 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, a semidirect factor, metacyclic, monomial, or rational 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: | $S_3^4:S_3$ |
| Order: | \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $S_3^5.D_4$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3\times C_3^5.C_3^5.C_6^2.C_2^4$, of order \(102036672\)\(\medspace = 2^{6} \cdot 3^{13} \) |
| $\operatorname{Aut}(H)$ | Group of order \(134\!\cdots\!240\)\(\medspace = 2^{14} \cdot 3^{21} \cdot 5 \cdot 7 \cdot 11^{2} \cdot 13^{2} \cdot 1093 \) |
| $\card{W}$ | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_3^{10}$ |
| 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 | not computed |