Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(1512\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 7 \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(10,11)(12,13), (10,12)(11,13)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), a semidirect factor, abelian (hence metabelian and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $A_4\times \SL(2,8)$ |
| Order: | \(6048\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 7 \) |
| Exponent: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, an A-group, and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_3\times \SL(2,8)$ |
| Order: | \(1512\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 7 \) |
| Exponent: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Automorphism Group: | $\SL(2,8):C_6$, of order \(3024\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 7 \) |
| Outer Automorphisms: | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $1$ |
The quotient is nonabelian, an A-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_4\times {}^2G(2,3)$, of order \(36288\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_3$, of order \(3\) |
Related subgroups
Other information
| Möbius function | $504$ |
| Projective image | $A_4\times \SL(2,8)$ |