Properties

Label 4805.a.155.a1.y1
Order $ 31 $
Index $ 5 \cdot 31 $
Normal Yes

Downloads

Learn more

Subgroup ($H$) information

Description:$C_{31}$
Order: \(31\)
Index: \(155\)\(\medspace = 5 \cdot 31 \)
Exponent: \(31\)
Generators: $ab^{20}$ Copy content Toggle raw display
Nilpotency class: $1$
Derived length: $1$

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

Ambient group ($G$) information

Description: $C_{31}\times C_{155}$
Order: \(4805\)\(\medspace = 5 \cdot 31^{2} \)
Exponent: \(155\)\(\medspace = 5 \cdot 31 \)
Nilpotency class:$1$
Derived length:$1$

The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 31$ (hence hyperelementary), and metacyclic.

Quotient group ($Q$) structure

Description: $C_{155}$
Order: \(155\)\(\medspace = 5 \cdot 31 \)
Exponent: \(155\)\(\medspace = 5 \cdot 31 \)
Automorphism Group: $C_2\times C_{60}$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Outer Automorphisms: $C_2\times C_{60}$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Nilpotency class: $1$
Derived length: $1$

The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 5,31$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

Automorphism information

While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.

$\operatorname{Aut}(G)$$C_4\times C_{30}.\PSL(2,31).C_2$
$\operatorname{Aut}(H)$ $C_{30}$, of order \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \)
$\operatorname{res}(S)$$C_{30}$, of order \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \)
$\card{\operatorname{ker}(\operatorname{res})}$\(3720\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
$W$$C_1$, of order $1$

Related subgroups

Centralizer:$C_{31}\times C_{155}$
Normalizer:$C_{31}\times C_{155}$
Complements:$C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$ $C_{155}$
Minimal over-subgroups:$C_{31}^2$$C_{155}$
Maximal under-subgroups:$C_1$
Autjugate subgroups:4805.a.155.a1.a14805.a.155.a1.b14805.a.155.a1.c14805.a.155.a1.d14805.a.155.a1.e14805.a.155.a1.f14805.a.155.a1.g14805.a.155.a1.h14805.a.155.a1.i14805.a.155.a1.j14805.a.155.a1.k14805.a.155.a1.l14805.a.155.a1.m14805.a.155.a1.n14805.a.155.a1.o14805.a.155.a1.p14805.a.155.a1.q14805.a.155.a1.r14805.a.155.a1.s14805.a.155.a1.t14805.a.155.a1.u14805.a.155.a1.v14805.a.155.a1.w14805.a.155.a1.x14805.a.155.a1.z14805.a.155.a1.ba14805.a.155.a1.bb14805.a.155.a1.bc14805.a.155.a1.bd14805.a.155.a1.be14805.a.155.a1.bf1

Other information

Möbius function$1$
Projective image$C_{155}$