Subgroup ($H$) information
| Description: | $C_2^8$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Index: | \(3024\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 7 \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(11,12)(13,14), (1,2)(13,14), (13,14)(17,18), (7,8)(13,14), (15,16)(17,18), (5,6)(17,18), (3,4)(17,18), (9,10)(13,14)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_2^9.{}^2G(2,3)$ |
| Order: | \(774144\)\(\medspace = 2^{12} \cdot 3^{3} \cdot 7 \) |
| 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: | $\SL(2,8):C_6$ |
| Order: | \(3024\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 7 \) |
| Exponent: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Automorphism Group: | ${}^2G(2,3)$, of order \(1512\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 7 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8:{}^2G(2,3)$, of order \(387072\)\(\medspace = 2^{11} \cdot 3^{3} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $\GL(8,2)$, of order \(5348063769211699200\)\(\medspace = 2^{28} \cdot 3^{5} \cdot 5^{2} \cdot 7^{2} \cdot 17 \cdot 31 \cdot 127 \) |
| $W$ | ${}^2G(2,3)$, of order \(1512\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 7 \) |
Related subgroups
| Centralizer: | $C_2^9$ |
| Normalizer: | $C_2^9.{}^2G(2,3)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^9.{}^2G(2,3)$ |