The 3-Sylow of the triple cover of the third Janko group

729.57 aka $MC(3)$ aka $3\text{-}\mathrm{Syl}(3.J_3)$

The 3-Sylow subgroup of the triple cover of $J_3$ turns out to have exciting and unexpected properties.

I will go over some context first.

$3.J_3$

The third Janko group has a mysterious relationship with threes. Its entry in ATLAS of Finite Group Representations shows that it’s triple cover has a $9$-dimensional representation (which is in fact unitary) over $\mathbb{F}_4$ , while the smallest representation of the “uncovered” group requires at least $18$ dimensions (and lives in $\mathbb{F}_9$, naturally).

ATLAS also lists its maximal subgroups, though without their orders. Thankfully, another resource has the full subgroup lattice of $J_3.$

No.	Subgroup	Order	Index	links
1	$L_2(16) : 2$	$8160$	$6156$	lmfdb atlas
2	$L_2(19)$	$3420$	$14688$	lmfdb atlas
3	$L_2(19)$	$3240$	$14688$	lmfdb atlas
4	$2^4 : (3 \times A_5)$	$2880$	$17442$	lmfdb
5	$L_2(17)$	$2448$	$20520$	lmfdb atlas
6	$(3 \times A_6):2_2$	$2160$	$23256$	lmfdb
7	$3^{2+1+2} : 8$	$1944$	$25840$	lmfdb	normalizer of the 3-Sylow
8	$2_{-}^{1 + 4} : A_5$	$1920$	$26163$	lmfdb	centralizer of involution
9	$2^{2 + 4} : (3 \times S_3)$	$1152$	$43605$	lmfdb
For all maximal subgroups except 7, the extension induced by going to the triple cover corresponds to a direct product with $C_3.$ For 7, the extension is non-split.

2-Sylow

• 2-Sylow subgroup: C₄².₁₅D⁴ This group was discovered as one of the two sporadic simple groups with a centralizer of an involution $2_{-}^{1 + 4} : A_5.$ Since the 2-Sylow is contained in the maximal subgroup 8, the sibling groups share their 2-Sylow.

This is also the 2-Sylow of one of extensions of $L_3(4)$ by an involution. Specifically, it’s 40320.o, which ATLAS calls $L_3(4):2a.$

TODO: more details

3-Sylow

TODO