Properties

Label 37056.a.1.a1.a1
Order $ 2^{6} \cdot 3 \cdot 193 $
Index $ 1 $
Normal Yes

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Subgroup ($H$) information

Description:$F_{193}$
Order: \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \)
Index: $1$
Exponent: \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \)
Generators: $a^{24}, b, a^{96}, a^{48}, a^{12}, a^{3}, a^{64}, a^{6}$ Copy content Toggle raw display
Derived length: $2$

The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, and a Z-group (hence supersolvable, monomial, metacyclic, metabelian, and an A-group).

Ambient group ($G$) information

Description: $F_{193}$
Order: \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \)
Exponent: \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \)
Derived length:$2$

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

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)$$F_{193}$, of order \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \)
$\operatorname{Aut}(H)$ $F_{193}$, of order \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \)
$W$$F_{193}$, of order \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \)

Related subgroups

Centralizer:$C_1$
Normalizer:$F_{193}$
Complements:$C_1$
Maximal under-subgroups:$C_{193}:C_{96}$$C_{193}:C_{64}$$C_{192}$

Other information

Möbius function$1$
Projective image$F_{193}$