Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(9,12)(10,11), (9,11)(10,12)\rangle$
|
| 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: | $C_2^3.\GL(2,\mathbb{Z}/4)$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $(C_2^3\times C_6):C_4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Automorphism Group: | $C_2\wr C_2^2\times D_6$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Outer Automorphisms: | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4:D_4\times S_4$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(512\)\(\medspace = 2^{9} \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^4:D_4$ | ||||||
| Normalizer: | $C_2^3.\GL(2,\mathbb{Z}/4)$ | ||||||
| Complements: | $(C_2^3\times C_6):C_4$ | ||||||
| Minimal over-subgroups: | $A_4$ | $C_2^3$ | $C_2^3$ | $C_2^3$ | $C_2^3$ | $C_2^3$ | $C_2^3$ |
| Maximal under-subgroups: | $C_2$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_2^3.\GL(2,\mathbb{Z}/4)$ |