Normal subgroup $\He_3^2:(C_2\times D_4) \trianglelefteq C_3^3.S_3\wr C_2^2$

• Label: 139968.kc.12.B
• Order: $ 2^{4} \cdot 3^{6} $
• Index: $ 2^{2} \cdot 3 $
• Normal: Yes

Subgroup ($H$) information

Description:	$\He_3^2:(C_2\times D_4)$
Order:	\(11664\)\(\medspace = 2^{4} \cdot 3^{6} \)
Index:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Generators:	$\langle(13,27)(14,25)(15,26)(19,32)(20,33)(21,31), (4,29,18)(5,30,16)(6,28,17) \!\cdots\! \rangle$
Derived length:	$4$

The subgroup is characteristic (hence normal), nonabelian, and solvable. Whether it is monomial 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:	$D_6$
Order:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism Group:	$D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \)
Outer Automorphisms:	$C_2$, of order \(2\)
Derived length:	$2$

The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, an A-group, 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 \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \)
$\operatorname{Aut}(H)$	$\He_3^2:C_2\wr C_2^2$, of order \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \)
$W$	$\He_3^2:C_2\wr C_2^2$, of order \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \)

Related subgroups

Centralizer:	$C_3$
Normalizer:	$C_3^3.S_3\wr C_2^2$
Minimal over-subgroups:	$\He_3^2:(C_6\times D_4)$	$C_3^4:D_6\wr C_2$	$\He_3^2:D_4:C_2^2$	$\He_3^2:C_2^3:C_4$
Maximal under-subgroups:	$C_3^3.S_3^3$	$\He_3^2:C_2^3$	$\He_3^2:(C_2\times C_4)$	$\He_3^2:D_4$	$\He_3^2:D_4$	$S_3^2:S_3^2$

Other information

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