Subgroup ($H$) information
| Description: | $C_3^3:D_6$ |
| Order: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Index: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(9,11,10)(12,14,15)(13,18,17), (1,8)(2,7)(3,6)(4,5)(13,17,18), (13,18,17), (10,11)(14,15)(17,18)(19,20), (12,14,15)(13,17,18), (16,19,20)\rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational.
Ambient group ($G$) information
| Description: | $C_6^4:(C_2\times S_4)$ |
| Order: | \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^3:S_4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2^4:S_4$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $3$ |
The quotient is nonabelian, monomial (hence solvable), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4:D_4$, of order \(497664\)\(\medspace = 2^{11} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_3:(C_3^3:C_2).\PSL(4,3).C_2$ |
| $\card{W}$ | \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
Related subgroups
| Centralizer: | $C_2^4$ | |
| Normalizer: | $C_6^4:(C_2\times S_4)$ | |
| Minimal over-subgroups: | $C_2\times C_3^4:C_6$ | |
| Maximal under-subgroups: | $C_3^3\times C_6$ | $C_3^3:S_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_6^3:(S_3\times S_4)$ |