Subgroup ($H$) information
| Description: | $C_3^7.C_3^5:Q_8$ |
| Order: | \(4251528\)\(\medspace = 2^{3} \cdot 3^{12} \) |
| Index: | \(3\) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(25,26,27)(28,30,29)(31,33,32), (4,6,5)(10,12,11)(13,15,14)(19,21,20)(25,26,27) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, and solvable. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^8.C_3^4:\SL(2,3)$ |
| Order: | \(12754584\)\(\medspace = 2^{3} \cdot 3^{13} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(459165024\)\(\medspace = 2^{5} \cdot 3^{15} \) |
| $\operatorname{Aut}(H)$ | Group of order \(1377495072\)\(\medspace = 2^{5} \cdot 3^{16} \) |
| $W$ | $C_3^8.C_3^4:\SL(2,3)$, of order \(12754584\)\(\medspace = 2^{3} \cdot 3^{13} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^8.C_3^4:\SL(2,3)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^8.C_3^4:\SL(2,3)$ |