Subgroup ($H$) information
| Description: | $C_3^6.D_{18}$ |
| Order: | \(26244\)\(\medspace = 2^{2} \cdot 3^{8} \) |
| Index: | \(2\) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\langle(1,16,7)(4,14,5)(6,13,15), (19,23,26)(20,25,24)(21,27,22), (10,11,18), (2,12,9) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, and supersolvable (hence solvable and monomial). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_3^5:S_3.D_{18}$ |
| Order: | \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| 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: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^7.C_3.C_6.C_6^2.C_2$, of order \(2834352\)\(\medspace = 2^{4} \cdot 3^{11} \) |
| $\operatorname{Aut}(H)$ | $\GL(2,4):C_2^4$, of order \(25509168\)\(\medspace = 2^{4} \cdot 3^{13} \) |
| $W$ | $C_3^5:S_3.D_{18}$, of order \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^5:S_3.D_{18}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^5:S_3.D_{18}$ |