Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(10,11)(12,13), (10,13)(11,12)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $C_3^5:(C_2^2\times S_4)$ |
| Order: | \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| 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: | $C_3^3.S_3^3$ |
| Order: | \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Automorphism Group: | $C_3^4.S_3^3$, of order \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| Outer Automorphisms: | $C_3$, of order \(3\) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3\times C_6^2).C_3^4.C_2^3$ |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_3^5.C_2^4$ | |
| Normalizer: | $C_3^5:(C_2^2\times S_4)$ | |
| Complements: | $C_3^3.S_3^3$ | |
| Minimal over-subgroups: | $C_2\times C_6$ | $C_2\times C_6$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^5:(C_2^2\times S_4)$ |