Properties

Label 1032.6.344.a1.a1
Order $ 3 $
Index $ 2^{3} \cdot 43 $
Normal Yes

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Subgroup ($H$) information

Description:$C_3$
Order: \(3\)
Index: \(344\)\(\medspace = 2^{3} \cdot 43 \)
Exponent: \(3\)
Generators: $a^{344}$ Copy content Toggle raw display
Nilpotency class: $1$
Derived length: $1$

The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $3$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.

Ambient group ($G$) information

Description: $C_{1032}$
Order: \(1032\)\(\medspace = 2^{3} \cdot 3 \cdot 43 \)
Exponent: \(1032\)\(\medspace = 2^{3} \cdot 3 \cdot 43 \)
Nilpotency class:$1$
Derived length:$1$

The ambient group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,43$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

Quotient group ($Q$) structure

Description: $C_{344}$
Order: \(344\)\(\medspace = 2^{3} \cdot 43 \)
Exponent: \(344\)\(\medspace = 2^{3} \cdot 43 \)
Automorphism Group: $C_2^2\times C_{42}$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Outer Automorphisms: $C_2^2\times C_{42}$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Nilpotency class: $1$
Derived length: $1$

The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,43$), hyperelementary, metacyclic, metabelian, a Z-group, 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_2^3\times C_{42}$, of order \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \)
$\operatorname{Aut}(H)$ $C_2$, of order \(2\)
$\operatorname{res}(\operatorname{Aut}(G))$$C_2$, of order \(2\)
$\card{\operatorname{ker}(\operatorname{res})}$\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
$W$$C_1$, of order $1$

Related subgroups

Centralizer:$C_{1032}$
Normalizer:$C_{1032}$
Complements:$C_{344}$
Minimal over-subgroups:$C_{129}$$C_6$
Maximal under-subgroups:$C_1$

Other information

Möbius function$0$
Projective image$C_{344}$