Subgroup ($H$) information
| Description: | $C_3^4$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(7,9,8)(10,12,11)(13,14,15)(16,17,18)(22,23,24)(28,30,29), (1,3,2)(4,6,5) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the radical, the socle, a semidirect factor, abelian (hence metabelian and an A-group), and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $C_3^4:A_6.C_2$ |
| Order: | \(58320\)\(\medspace = 2^{4} \cdot 3^{6} \cdot 5 \) |
| Exponent: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $A_6.C_2$ |
| Order: | \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Automorphism Group: | $S_6:C_2$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $1$ |
The quotient is nonabelian, almost simple, and nonsolvable.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4:A_6.D_6$, of order \(349920\)\(\medspace = 2^{5} \cdot 3^{7} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2.\PSL(4,3).C_2$ |
| $W$ | $A_6.C_2$, of order \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_3^4$ | ||
| Normalizer: | $C_3^4:A_6.C_2$ | ||
| Complements: | $A_6.C_2$ $A_6.C_2$ $A_6.C_2$ | ||
| Minimal over-subgroups: | $C_3^4:C_5$ | $C_3^4:C_3$ | $C_3^2\wr C_2$ |
| Maximal under-subgroups: | $C_3^3$ | $C_3^3$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_3^4:A_6.C_2$ |