Properties

Label 448.652.8.b1.a1
Order $ 2^{3} \cdot 7 $
Index $ 2^{3} $
Normal Yes

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Subgroup ($H$) information

Description:$C_2\times C_{28}$
Order: \(56\)\(\medspace = 2^{3} \cdot 7 \)
Index: \(8\)\(\medspace = 2^{3} \)
Exponent: \(28\)\(\medspace = 2^{2} \cdot 7 \)
Generators: $a^{2}, c^{4}, c^{14}, b^{2}c^{14}$ Copy content Toggle raw display
Nilpotency class: $1$
Derived length: $1$

The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.

Ambient group ($G$) information

Description: $(C_2\times C_8).D_{14}$
Order: \(448\)\(\medspace = 2^{6} \cdot 7 \)
Exponent: \(56\)\(\medspace = 2^{3} \cdot 7 \)
Derived length:$2$

The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Quotient group ($Q$) structure

Description: $C_2^3$
Order: \(8\)\(\medspace = 2^{3} \)
Exponent: \(2\)
Automorphism Group: $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Outer Automorphisms: $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Nilpotency class: $1$
Derived length: $1$

The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), 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)$$C_7.(C_2^4\times C_6).C_2^5$
$\operatorname{Aut}(H)$ $C_6\times D_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \)
$\operatorname{res}(\operatorname{Aut}(G))$$C_2^2\times C_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \)
$\card{\operatorname{ker}(\operatorname{res})}$\(896\)\(\medspace = 2^{7} \cdot 7 \)
$W$$C_2$, of order \(2\)

Related subgroups

Centralizer:$C_{14}\times \OD_{16}$
Normalizer:$(C_2\times C_8).D_{14}$
Minimal over-subgroups:$C_2^2\times C_{28}$$C_{28}:C_4$$C_{28}:C_4$$C_2\times C_{56}$$C_2\times C_{56}$$C_{14}:C_8$$C_{14}:C_8$
Maximal under-subgroups:$C_2\times C_{14}$$C_{28}$$C_{28}$$C_2\times C_4$

Other information

Möbius function$-8$
Projective image$C_2\times D_{14}$