Subgroup ($H$) information
| Description: | $S_3\times {}^2G(2,3)$ |
| Order: | \(9072\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 7 \) |
| Index: | $1$ |
| Exponent: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Generators: |
$\langle(11,12), (1,9,8)(2,3,7)(4,6,5)(10,12,11), (1,3,2)(4,9,5)(10,11), (10,11,12)\rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a direct factor, nonabelian, a Hall subgroup, 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: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 \(9072\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $S_3\times {}^2G(2,3)$, of order \(9072\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 7 \) |
| $W$ | $S_3\times {}^2G(2,3)$, of order \(9072\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 7 \) |
Related subgroups
| Centralizer: | $C_1$ | |||||
| Normalizer: | $S_3\times {}^2G(2,3)$ | |||||
| Complements: | $C_1$ | |||||
| Maximal under-subgroups: | $C_3\times {}^2G(2,3)$ | $S_3\times \SL(2,8)$ | $\SL(2,8):C_6$ | $F_8:C_3\times S_3$ | $C_3^2.S_3^2$ | $S_3\times F_7$ |
Other information
| Möbius function | $1$ |
| Projective image | $S_3\times {}^2G(2,3)$ |