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⋉

left normal factor semidirect product

Left Semidirect Product Operator

⋉ marks a semidirect product, where one group acts on another. In the common group-theory convention, H ⋉ N means H acts on N, the normal subgroup on the right. Some sources reverse the convention, so use the definition given with the formula.

Text symbol

Also called: ltimes · semidirect product
Search it as: H ⋉ N · normal subgroup · group action · LaTeX ltimes
Shows up in: Anything that takes text · its shape is whatever your font draws

What ⋉ means

It marks a semidirect product, where one group acts on another.

For H ⋉ N, H is the acting group and N is a normal subgroup: conjugating N by elements of the whole product keeps it inside N. That role distinction is why swapping the factors can change the construction.

Unlike a direct product, the factors do not simply commute: the acting group can change elements of the normal group.

Which factor is normal in H ⋉ N?

In the common convention, N on the right is normal.

Unicode's character names call ⋉ the left-normal-factor form and ⋊ the right-normal-factor form; mathematical formula conventions do not always follow those names. Both conventions appear in mathematical texts as of 2026, so follow the local definition.

Technical group-theory notation whose meaning depends on the definition surrounding the formula.

Do not assume the left factor is normal; in common notation H ⋉ N, N is normal.

What it means when someone sends it to you

3 readings

H ⋉ N, common convention

H acts on N; N, the right-hand factor, is normal.

N ⋉ H, common convention

N acts on H; H, the right-hand factor, is normal.

N ⋉ H, reversed convention

H acts on N; N is normal on the left.

Don't confuse it with

⋊

Right Normal Factor Semidirect Product

In common group notation, N ⋊ H gives the same roles as H ⋉ N; the reversed shape can make formulas easy to swap.

Combination ideas

Common convention: H acts on N, the normal factor.

Common convention: N acts on H, which is normal.

Common convention: N is normal and H acts on it.

Questions people ask

Why does ⋉ look different in my document?

It can. As of 2026, Unicode says its reference glyphs are not prescriptive, so mathematical fonts may draw ⋉ with different shapes and spacing.

How to type ⋉

Windows (Word)

Alt+XType 22C9, then Alt+X to get ⋉.

Windows

Win+Rrun charmap for the full table, or use the emoji picker's Ω symbols tab (Win + .).

macOS

Control+Command+SpaceOpen the character viewer and search “left normal factor semidirect product”.

Linux

Ctrl+Shift+UType 22c9, then Enter or Space.

HTML

⋉

CSS

\22C9

LaTeX

As of 2026, load AMS math symbols and write `\ltimes` in math mode.

Unicode

Insert U+22C9 through your editor's Unicode input feature if it supports mathematical symbols.

Codes and encoding1 code points

HTML decimal

⋉

HTML hex

⋉

CSS content

content: "\22C9";

Each escape is closed by the next backslash; a trailing space is needed before a literal hex digit.

URL encoded

%E2%8B%89

On screen

Code points

JS .length

UTF-8 bytes

A visible symbol can contain several code points. JS .length counts UTF-16 units; UTF-8 counts bytes.

Families this belongs to

Maths & numbers1,835 symbols

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