Subgroup ($H$) information
| Description: | not computed |
| Order: | \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | not computed |
| Generators: |
$b^{2}c^{3}de^{3}fg, e^{3}f^{3}, c^{2}d^{4}e^{5}, f^{2}, f^{3}, d^{2}f^{3}g^{2}, c^{4}d^{2}e^{4}fg, d^{3}, c^{3}, e^{2}g$
|
| Derived length: | not computed |
The subgroup is the commutator subgroup (hence characteristic and normal), a semidirect factor, nonabelian, and metabelian (hence solvable). Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_2^5:(\He_3^2:C_4)$ |
| Order: | \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2\times C_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^{12}.C_2^5.A_4$, of order \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $C_3^{12}.C_2^6.D_6.(C_2\times D_4)$, of order \(6530347008\)\(\medspace = 2^{12} \cdot 3^{13} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |