Subgroup ($H$) information
| Description: | ${}^2G(2,3)$ |
| 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: |
$\langle(1,9,8)(2,3,7)(4,6,5), (1,3,2)(4,9,5)\rangle$
|
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, nonabelian, almost simple, 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: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times {}^2G(2,3)$, of order \(9072\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | ${}^2G(2,3)$, of order \(1512\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 7 \) |
| $W$ | ${}^2G(2,3)$, of order \(1512\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 7 \) |
Related subgroups
| Centralizer: | $S_3$ | |||
| Normalizer: | $S_3\times {}^2G(2,3)$ | |||
| Complements: | $S_3$ $S_3$ $S_3$ | |||
| Minimal over-subgroups: | $C_3\times {}^2G(2,3)$ | $\SL(2,8):C_6$ | ||
| Maximal under-subgroups: | $\SL(2,8)$ | $F_8:C_3$ | $C_9:C_6$ | $F_7$ |
Other information
| Möbius function | $3$ |
| Projective image | $S_3\times {}^2G(2,3)$ |