Subgroup ($H$) information
| Description: | $C_1$ |
| Order: | $1$ |
| Index: | \(508032\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 7^{2} \) |
| Exponent: | $1$ |
| Generators: | |
| Nilpotency class: | $0$ |
| Derived length: | $0$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), the Frattini subgroup, the Fitting subgroup, the radical, a direct factor, cyclic (hence elementary (for every $p$), hyperelementary, metacyclic, and a Z-group), stem, a $p$-group (for every $p$), perfect, and rational.
Ambient group ($G$) information
| Description: | $\SOPlus(4,8)$ |
| Order: | \(508032\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 7^{2} \) |
| Exponent: | \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $\SOPlus(4,8)$ |
| Order: | \(508032\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 7^{2} \) |
| Exponent: | \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Automorphism Group: | $\SL(2,8)^2:C_6$, of order \(1524096\)\(\medspace = 2^{7} \cdot 3^{5} \cdot 7^{2} \) |
| Outer Automorphisms: | $C_3$, of order \(3\) |
| Nilpotency class: | $-1$ |
| Derived length: | $1$ |
The quotient is nonabelian 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\SL(2,8)^2:C_6$, of order \(1524096\)\(\medspace = 2^{7} \cdot 3^{5} \cdot 7^{2} \) |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $\SOPlus(4,8)$ | |||||||
| Normalizer: | $\SOPlus(4,8)$ | |||||||
| Complements: | $\SOPlus(4,8)$ | |||||||
| Minimal over-subgroups: | $C_7$ | $C_7$ | $C_7$ | $C_3$ | $C_3$ | $C_2$ | $C_2$ | $C_2$ |
Other information
| Möbius function | $0$ |
| Projective image | $\SOPlus(4,8)$ |