Subgroup ($H$) information
| Description: | $C_2^3$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(559872\)\(\medspace = 2^{8} \cdot 3^{7} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(13,14)(15,16)(17,18)(19,20)(21,22)(23,24), (7,8)(9,10)(11,12)(13,14)(15,16) \!\cdots\! \rangle$
|
| 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: | $C_3^6.(C_2^9.D_6)$ |
| Order: | \(4478976\)\(\medspace = 2^{11} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_3^6.C_2^5:S_4$ |
| Order: | \(559872\)\(\medspace = 2^{8} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Automorphism Group: | $C_3^6.C_2^7:S_4$, of order \(2239488\)\(\medspace = 2^{10} \cdot 3^{7} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $4$ |
The quotient is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_2^6.A_4.C_2^6.C_2^2$, of order \(143327232\)\(\medspace = 2^{16} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_3^6.C_2^6.C_2^4$ |
| Normalizer: | $C_3^6.(C_2^9.D_6)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^6.C_2^8:D_6$ |