Subgroup ($H$) information
| Description: | $C_3^5:D_{18}$ |
| Order: | \(8748\)\(\medspace = 2^{2} \cdot 3^{7} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$a^{3}c^{3}d^{2}e, de^{2}f^{2}g, eg, g, b^{9}cd^{2}e^{2}f^{2}g, b^{8}c^{4}d^{2}efg^{2}, c^{2}, b^{6}, fg^{2}$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), 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: | $C_3^5.S_3^3$ |
| Order: | \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Quotient group ($Q$) structure
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4:(C_4^2:C_2^2)$, of order \(209952\)\(\medspace = 2^{5} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_3^5.C_3^3.C_6.C_2^3$, of order \(629856\)\(\medspace = 2^{5} \cdot 3^{9} \) |
| $W$ | $C_3^4.S_3^3$, of order \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^5.S_3^3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^5.S_3^3$ |