Subgroup ($H$) information
| Description: | $C_2^2:A_4$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Index: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$ac^{2}e^{2}, e^{3}, bc^{3}, c^{3}, d^{3}e^{3}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_3^2.A_4^2$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and metabelian (hence solvable). Whether it is monomial has not been computed.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4.C_3^4.S_3^3.C_2$ |
| $\operatorname{Aut}(H)$ | $\AGammaL(2,4)$, of order \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{res}(S)$ | $\POPlus(4,3)$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| $W$ | $C_2^2:A_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_3^2$ | ||
| Normalizer: | $C_6^2:A_4$ | ||
| Normal closure: | $C_2^4:C_3^2$ | ||
| Core: | $C_2^4$ | ||
| Minimal over-subgroups: | $C_2^4:C_3^2$ | $C_2^4:C_3^2$ | |
| Maximal under-subgroups: | $C_2^4$ | $A_4$ | $A_4$ |
Other information
| Number of subgroups in this autjugacy class | $18$ |
| Number of conjugacy classes in this autjugacy class | $6$ |
| Möbius function | $0$ |
| Projective image | $C_3^2.A_4^2$ |