Subgroup ($H$) information
| Description: | $D_6\times C_2^4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(3\) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,3)(2,4)(5,7)(6,8), (1,4)(2,3), (1,4)(2,3)(5,8)(6,7), (12,13), (10,11)(12,13), (1,2)(3,4), (9,10,11)\rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, an A-group, and rational.
Ambient group ($G$) information
| Description: | $C_2^2:A_4\times D_6$ |
| Order: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times C_2^4.(C_6\times A_5).C_2$ |
| $\operatorname{Aut}(H)$ | $C_2^5.\GL(5,2)\times S_3$, of order \(1919877120\)\(\medspace = 2^{16} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 31 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_3\times C_6:S_5$, of order \(4320\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $C_3\times S_3$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $S_3\times C_2^2:A_4$ |