Properties

Label 7200.dx.4.a1.a1
Order $ 2^{3} \cdot 3^{2} \cdot 5^{2} $
Index $ 2^{2} $
Normal Yes

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Subgroup ($H$) information

Description:$C_5\times A_6$
Order: \(1800\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Index: \(4\)\(\medspace = 2^{2} \)
Exponent: \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Generators: $\langle(1,3,4,2,5)(6,15)(7,12)(8,9)(11,13), (1,5,2,4,3)(6,10,9,7)(12,15,13,14), (1,2,3,5,4)\rangle$ Copy content Toggle raw display
Derived length: $1$

The subgroup is the socle (hence characteristic and normal), nonabelian, and nonsolvable.

Ambient group ($G$) information

Description: $S_6:C_{10}$
Order: \(7200\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Exponent: \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Derived length:$1$

The ambient group is nonabelian and nonsolvable.

Quotient group ($Q$) structure

Description: $C_2^2$
Order: \(4\)\(\medspace = 2^{2} \)
Exponent: \(2\)
Automorphism Group: $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \)
Outer Automorphisms: $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \)
Derived length: $1$

The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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_4\times S_6:C_2$, of order \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \)
$\operatorname{Aut}(H)$ $C_4\times S_6:C_2$, of order \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \)
$W$$S_6:C_2$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \)

Related subgroups

Centralizer:$C_5$
Normalizer:$S_6:C_{10}$
Minimal over-subgroups:$C_5\times S_6$$C_5\times \PGL(2,9)$$A_6.C_{10}$
Maximal under-subgroups:$A_6$$C_5\times A_5$$C_3^2:C_{20}$$C_5\times S_4$

Other information

Möbius function$2$
Projective image$S_6:C_2$