Subgroup ($H$) information
| Description: | $M_{12}$ |
| Order: | \(95040\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| Index: | $1$ |
| Exponent: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$\langle(2,9,4)(3,12,6)(7,8,11), (1,12)(2,6)(3,8)(4,9)(5,7)(10,11)\rangle$
|
| Derived length: | $0$ |
The subgroup is the commutator subgroup (hence characteristic and normal), the socle, a direct factor, nonabelian, a Hall subgroup, and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
Ambient group ($G$) information
| Description: | $M_{12}$ |
| Order: | \(95040\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| Exponent: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $M_{12}:C_2$, of order \(190080\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $M_{12}:C_2$, of order \(190080\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| $W$ | $M_{12}$, of order \(95040\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \cdot 11 \) |
Related subgroups
Other information
| Möbius function | $1$ |
| Projective image | $M_{12}$ |