Subgroup ($H$) information
| Description: | $C_3^4$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(2,6,7), (4,11,5), (3,9,12)(4,11,5), (1,8,10)\rangle$
|
| 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), and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $C_3^2:S_3\wr D_4$ |
| Order: | \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_{12}^2:D_4$ |
| Order: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Automorphism Group: | $C_3:(C_2^4.C_2^5.C_2^2)$ |
| Outer Automorphisms: | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian and supersolvable (hence solvable and monomial).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^7:D_4$, of order \(248832\)\(\medspace = 2^{10} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $C_2.\PSL(4,3).C_2$ |
| $W$ | $C_2\wr D_4$, of order \(128\)\(\medspace = 2^{7} \) |
Related subgroups
| Centralizer: | $C_3^6$ | ||
| Normalizer: | $C_3^2:S_3\wr D_4$ | ||
| Complements: | $C_{12}^2:D_4$ | ||
| Minimal over-subgroups: | $C_3^5$ | $C_3^5$ | $C_3^3:S_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^2:S_3\wr D_4$ |