Subgroup ($H$) information
| Description: | $C_3^7$ |
| Order: | \(2187\)\(\medspace = 3^{7} \) |
| Index: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(19,20,21)(22,23,24)(31,32,33)(34,35,36), (16,17,18)(19,20,21)(28,30,29) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary). Whether it is a direct factor, a semidirect factor, metacyclic, monomial, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^7.C_3:S_3^3$ |
| Order: | \(1417176\)\(\medspace = 2^{3} \cdot 3^{11} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Quotient group ($Q$) structure
| Description: | $C_3:S_3^3$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3\wr S_4$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \) |
| Outer Automorphisms: | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^8.C_3^4.C_6^3.C_2^3$, of order \(918330048\)\(\medspace = 2^{6} \cdot 3^{15} \) |
| $\operatorname{Aut}(H)$ | Group of order \(134\!\cdots\!240\)\(\medspace = 2^{14} \cdot 3^{21} \cdot 5 \cdot 7 \cdot 11^{2} \cdot 13^{2} \cdot 1093 \) |
| $W$ | $C_3:S_3^3$, of order \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_3^7$ |
| Normalizer: | $C_3^7.C_3:S_3^3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^7.C_3:S_3^3$ |