Subgroup ($H$) information
| Description: | $C_2^4:A_4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(8,10)(9,13)(11,15)(12,14), (4,5)(6,7)(8,10)(11,15), (4,5,6)(8,10,15)(12,13,14) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is the commutator subgroup (hence characteristic and normal), a semidirect factor, nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_3\times C_2^5:S_4$ |
| Order: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_2\times C_6$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Outer Automorphisms: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.C_6.C_2^5$, of order \(12288\)\(\medspace = 2^{12} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $\AGammaL(3,4)$, of order \(23224320\)\(\medspace = 2^{13} \cdot 3^{4} \cdot 5 \cdot 7 \) |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_2^5:S_4$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3\times C_2^5:S_4$ |