Subgroup ($H$) information
| Description: | $C_2^5$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(1,2)(3,8)(4,6)(5,7), (1,3)(2,8)(4,5)(6,7), (1,4)(2,6)(3,5)(7,8)(9,12)(10,11), (1,5)(2,7)(3,4)(6,8), (1,5)(2,7)(3,4)(6,8)(9,11)(10,12)\rangle$
|
| 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), and rational.
Ambient group ($G$) information
| Description: | $C_2^5:A_4$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), and metabelian.
Quotient group ($Q$) structure
| Description: | $A_4$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.C_2^6.C_3.S_3$, of order \(73728\)\(\medspace = 2^{13} \cdot 3^{2} \) |
| $\operatorname{Aut}(H)$ | $\GL(5,2)$, of order \(9999360\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $\PSOPlus(4,3)$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(256\)\(\medspace = 2^{8} \) |
| $W$ | $A_4$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^5$ | |||
| Normalizer: | $C_2^5:A_4$ | |||
| Complements: | $A_4$ | |||
| Minimal over-subgroups: | $C_2^3:A_4$ | $D_4\times C_2^3$ | ||
| Maximal under-subgroups: | $C_2^4$ | $C_2^4$ | $C_2^4$ | $C_2^4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $4$ |
| Projective image | $C_2^5:A_4$ |