Subgroup ($H$) information
| Description: | $C_3^2\wr C_3$ |
| Order: | \(2187\)\(\medspace = 3^{7} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$\langle(10,14,15)(11,12,16)(13,17,18)(19,23,24)(20,21,25)(22,26,27), (10,12,17) \!\cdots\! \rangle$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), a semidirect factor, nonabelian, a $3$-Sylow subgroup (hence a Hall subgroup), a $p$-group (hence elementary and hyperelementary), and metabelian. Whether it is a direct factor has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6.(S_3\times Q_8)$ |
| Order: | \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2\times Q_8$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2^3:S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Outer Automorphisms: | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.Q_8.C_3^3.C_2^2$, of order \(629856\)\(\medspace = 2^{5} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_3^7.C_3^3.Q_8.C_3^3.C_2^2$, of order \(51018336\)\(\medspace = 2^{5} \cdot 3^{13} \) |
| $W$ | $C_{10}^2:C_{15}:C_4$, of order \(6000\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{3} \) |
Related subgroups
| Centralizer: | $C_3^2$ | ||
| Normalizer: | $C_3^6.(S_3\times Q_8)$ | ||
| Minimal over-subgroups: | $C_3^4.C_3^3.C_2$ | $C_3^6.S_3$ | $C_3^6.C_6$ |
| Maximal under-subgroups: | $C_3^6$ | $C_3^5:C_3$ | $C_3^5.C_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_3^6.(S_3\times Q_8)$ |