Normal subgroup $C_3^4.C_3^3.C_2^2 \trianglelefteq C_3^3.S_3\wr C_2^2$

• Label: 139968.kc.16.A
• Order: $ 2^{2} \cdot 3^{7} $
• Index: $ 2^{4} $
• Normal: Yes

Subgroup ($H$) information

Description:	not computed
Order:	\(8748\)\(\medspace = 2^{2} \cdot 3^{7} \)
Index:	\(16\)\(\medspace = 2^{4} \)
Exponent:	not computed
Generators:	$\langle(13,27)(14,25)(15,26)(19,32)(20,33)(21,31), (1,3,2)(7,20,31)(8,21,32)(9,19,33) \!\cdots\! \rangle$
Derived length:	not computed

The subgroup is characteristic (hence normal), nonabelian, and supersolvable (hence solvable and monomial). Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.

Ambient group ($G$) information

Description:	$C_3^3.S_3\wr C_2^2$
Order:	\(139968\)\(\medspace = 2^{6} \cdot 3^{7} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Derived length:	$4$

The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.

Quotient group ($Q$) structure

Description:	$C_2\times D_4$
Order:	\(16\)\(\medspace = 2^{4} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism Group:	$C_2\wr C_2^2$, of order \(64\)\(\medspace = 2^{6} \)
Outer Automorphisms:	$C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \)
Derived length:	$2$

The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, 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 $W = G / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_3^2.C_3^5.C_2^3.C_2^4$, of order \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \)
$\operatorname{Aut}(H)$	not computed
$W$	$S_3\wr C_2^2$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \)

Related subgroups

Centralizer:	$C_3^3$
Normalizer:	$C_3^3.S_3\wr C_2^2$
Minimal over-subgroups:	$C_3^3.C_3^4.C_2^3$	$C_3^4.S_3^3$	$C_3^3.C_3^4.C_2^2.C_2$	$C_3^4.C_3^3.C_2^3$	$C_3^4.C_3:S_3.C_6.C_2$	$C_3^4.C_3^3.D_4$	$C_3^4.C_3^3.C_2^2.C_2$
Maximal under-subgroups:	$C_3^4.C_3^3.C_2$	$C_3.C_3^5.C_6$	$C_3^4.S_3^2$	$C_3.C_3^5.C_2^2$	$C_3.C_3^5.C_2^2$

Other information

Number of conjugacy classes in this autjugacy class	$1$
Möbius function	not computed
Projective image	$C_3^3.S_3\wr C_2^2$