Subgroup ($H$) information
| Description: | $C_3\times \SL(2,8)$ |
| Order: | \(1512\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 7 \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Generators: |
$\langle(1,7,5)(2,6,3)(4,8,9)(10,12,11), (1,6,3)(2,9,7)(4,8,5)(10,11,12), (10,11,12)\rangle$
|
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), the socle, a semidirect factor, nonabelian, an A-group, and nonsolvable.
Ambient group ($G$) information
| Description: | $S_3\times {}^2G(2,3)$ |
| Order: | \(9072\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 7 \) |
| Exponent: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times {}^2G(2,3)$, of order \(9072\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $\SL(2,8):C_6$, of order \(3024\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 7 \) |
| $W$ | $\SL(2,8):C_6$, of order \(3024\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 7 \) |
Related subgroups
Other information
| Möbius function | $1$ |
| Projective image | $S_3\times {}^2G(2,3)$ |