Properties

Label 1848.26.42.a1.a1
Order $ 2^{2} \cdot 11 $
Index $ 2 \cdot 3 \cdot 7 $
Normal Yes

Downloads

Learn more

Subgroup ($H$) information

Description:$C_{44}$
Order: \(44\)\(\medspace = 2^{2} \cdot 11 \)
Index: \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \)
Exponent: \(44\)\(\medspace = 2^{2} \cdot 11 \)
Generators: $b^{77}, b^{154}, b^{28}$ Copy content Toggle raw display
Nilpotency class: $1$
Derived length: $1$

The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a direct factor, and cyclic (hence elementary ($p = 2,11$), hyperelementary, metacyclic, and a Z-group).

Ambient group ($G$) information

Description: $F_7\times C_{44}$
Order: \(1848\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 11 \)
Exponent: \(924\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 11 \)
Derived length:$2$

The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.

Quotient group ($Q$) structure

Description: $F_7$
Order: \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \)
Exponent: \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \)
Automorphism Group: $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \)
Outer Automorphisms: $C_1$, of order $1$
Nilpotency class: $-1$
Derived length: $2$

The quotient is nonabelian and a Z-group (hence solvable, supersolvable, monomial, metacyclic, 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_2^2\times C_{10}\times F_7$, of order \(1680\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
$\operatorname{Aut}(H)$ $C_2\times C_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \)
$\operatorname{res}(\operatorname{Aut}(G))$$C_2\times C_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \)
$\card{\operatorname{ker}(\operatorname{res})}$\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
$W$$C_1$, of order $1$

Related subgroups

Centralizer:$F_7\times C_{44}$
Normalizer:$F_7\times C_{44}$
Complements:$F_7$ $F_7$
Minimal over-subgroups:$C_{308}$$C_{132}$$C_2\times C_{44}$
Maximal under-subgroups:$C_{22}$$C_4$

Other information

Möbius function$-7$
Projective image$F_7$