Some group multiplication tables

Multiplication tables for groups of small order

Multiplication tables for groups of order 2 through 10

The multiplication tables given below cover the groups of order 10 or less. That is, any group of order 2 through 10 is isomorphic to one of the groups given on this page. The reader needs to know these definitions: group, cyclic group, symmetric group, dihedral group, direct product of groups, subgroup, normal subgroup.

The quaternion group is discussed in Example 3.3.7. There are more group tables at the end of Section 7.10.

C₂, the cyclic group of order 2

Elements:
order 2: a

Subgroups:
order 2: {1,a}
order 1: {1}

C₃, the cyclic group of order 3

Elements:
order 3: a, a²

Subgroups:
order 3: {1,a,a²}
order 1: {1}

C₄, the cyclic group of order 4

Elements:
order 4: a, a³
order 2: a²

Subgroups:
order 4: {1,a,a²,a³}
order 2: {1,a²}
order 1: {1}

V, the Klein four group

Elements:
order 2: a, b, ab

Subgroups:
order 4: {1,a,b,ab}
order 2: {1,a}, {1,b}, {1,ab}
order 1: {1}
 

C₅, the cyclic group of order 5

Elements:
order 5: a, a², a³, a⁴

Subgroups:
order 5: {1,a,a²,a³,a⁴}
order 1: {1}

C₆, the cyclic group of order 6

Elements:
order 6: a, a⁵
order 3: a², a⁴
order 2: a³

Subgroups:
order 6: {1,a,a²,a³,a⁴,a⁵}
order 3: {1,a²,a⁴}
order 2: {1,a³}
order 1: {1}

S₃, the symmetric group on three elements

Elements:
order 3: a, a²
order 2: b, ab, a²b

Subgroups:
order 6: {1,a,a²,b,ab,a²b}
order 3: {1,a,a²}
order 2: {1,b}, {1,ab}, {1,a²b}
order 1: {1}
 

Normal subgroups:
order 6: {1,a,a²,b,ab,a²b}
order 3: {1,a,a²}
order 1: {1}
 

C₇, the cyclic group of order 7

Elements:
order 7: a, a², a³, a⁴, a⁵, a⁶

Subgroups:
order 7: {1,a,a²,a³,a⁴, a⁵,a⁶}
order 1: {1}

C₈, the cyclic group of order 8

Elements:
order 8: a, a³, a⁵, a⁷
order 4: a², a⁶
order 2: a⁴

Subgroups:
order 8: {1,a,a²,a³,a⁴, a⁵, a⁶, a⁷}
order 4: {1,a²,a⁴,a⁶}
order 2: {1,a⁴}
order 1: {1}

C₄ x C₂, the direct product of a cyclic group of order 4 and a cyclic group of order 2

Elements:
order 4: a, a³, ab, a³b
order 2: a², b, a²b
order 1: 1

Subgroups:
order 8: {1,a,a²,a³, b,ab,a²b,a³b}
order 4: {1,a,a²,a³} {1,ab,a²,a³b} {1,a²,b,a²b}
order 2: {1,a²}, {1,b}, {1,a²b}
order 1: {1}

C₂ x C₂ x C₂, the direct product of 3 cyclic groups of order 2

Elements:
order 2: a, b, ab, c, ac, bc, abc

Subgroups:
order 8: { 1, a, b, ab, c, ac, bc, abc }
order 4: {1,a,b,ab}, {1,a,c,ac}, {1,a,bc,abc}, {1,b,c,bc}, {1,b,ac,abc}, {1,ab,c,abc}, {1,ab,ac,bc}
order 2: {1,a}, {1,b}, {1,ab}, {1,c}, {1,ac}, {1,bc}, {1,abc}
order 1: {1}

D₄, the dihedral group of order eight

Elements:
order 4: a, a³
order 2: a², b, ab, a²b, a³b

Subgroups:
order 8: {1,a,a²,a³, b,ab,a²b,a³b}
order 4: {1,a²,b,a²b}, {1,a,a²,a³}, {1,a²,ab,a³b}
order 2: {1,b}, {1,a²b}, {1,a²}, {1,ab}, {1,a³b}
order 1: {1}
 

Normal subgroups:
order 8: {1,a,a²,a³, b,ab,a²b,a³b}
order 4: {1,a²,b,a²b}, {1,a,a²,a³}, {1,a²,ab,a³b}
order 2: {1,a²}
order 1: {1}
 

Q, the quaternion group (of order eight)

Elements:
order 4: a, a³, b, ab, a²b, a³b
order 2: a²

Subgroups:
order 8: {1,a,a²,a³,b,ab,a²b,a³}
order 4: {1,a,a²,a³}, {1,b,a²,a²b}, {1,ab,a²,a³b}
order 2: {1,a²}
order 1: {1}
 

Normal subgroups:
order 8: {1,a,a²,a³,b,ab,a²b,a³}
order 4: {1,a,a²,a³}, {1,b,a²,a²b}, {1,ab,a²,a³b}
order 2: {1,a²}
order 1: {1}
 

Here are several different patterns for the multiplication table of the quaternion group, using the cross product of unit vectors i, j, k:

Elements:
order 4: i, -i, j, -j, k, -k
order 2: -1
 

Subgroups:
order 8: {1,-1,i,-i,j,-j,k,-k}
order 4: {1,i,-1,-i}, {1,j,-1,-j}, {1,k,-1,-k}
order 2: {1,-1}
order 1: {1}
 

Normal subgroups:
order 8: {1,-1,i,-i,j,-j,k,-k}
order 4: {1,i,-1,-i}, {1,j,-1,-j}, {1,k,-1,-k}
order 2: {1,-1}
order 1: {1}
 

C₉, the cyclic group of order 9

Elements:
order 9: a, a², a⁴, a⁵, a⁶, a⁷
order 3: a³, a⁶
 

Subgroups:
order 9: {1,a,a²,a³,a⁴, a⁵, a⁶, a⁷, a⁸}
order 3: {1,a³,a⁶}
order 1: {1}

C₃ x C₃, the direct product of two cyclic groups of order 3

Elements:
order 3: a, a², b, ab, a²b, b², ab², a²b²
 

Subgroups:
order 3: {1,a,a²}, {1,b,b²}, {1,ab,a²b²}, {1,a²b,ab²}
order 1: {1}

C₁₀, the cyclic group of order 10

Elements:
order 10: a, a³, a⁷, a⁹
order 5: a², a⁴, a⁶, a⁸
order 2: a⁵

Subgroups:
order 10: {1,a,a²,a³,a⁴, a⁵, a⁶, a⁷, a⁸, a⁹}
order 5: {1,a²,a⁴, a⁶,a⁸}
order 2: {1,a⁵}
order 1: {1}

D₅, the dihedral group of order ten

Elements:
order 5: a, a², a³, a⁴
order 2: b, ab, a²b, a³b, a⁴b
 

Subgroups:
order 10: {1,a,a²,a³,a⁴, b,ab,a²b,a³b,a⁴b}
order 5: {1,a,a²,a³,a⁴}
order 2: {1,b}, {1,ab} {1,a²b}, {1,a³b}, {1,a⁴b}
order 1: {1}
 

Normal subgroups:
order 10: {1,a,a²,a³,a⁴, b,ab,a²b,a³b,a⁴b}
order 5: {1,a,a²,a³,a⁴}
order 1: {1}