group Small finite groups and Cayley tables Explanations and comments GAPid Group Presentation 3_1 cyclic group C3 < a | a3 > 4_1 cyclic group C4 < a | a4 > 4_1b cyclic group C4 < i | i4 > 4_2 Klein group K4 < a,b | a2=b2=(ab)2 > 4_2a Klein group K4 (rectangle) < a,b | a2=b2=(ab)2 > 4_2b Klein group K4 (losange) < a,b | a2=b2=(ab)2 > 6_2a cyclic group C6 < a | a6 > 6_2b group C3xC2 < a,b | a3=b2=aba-1b > 6_1 symmetric group S3,dihedral Dih6 (triangle) < a,c | a3=c2=acac > 8_1 cyclic group C8 < a | a8 > 8_2 group C4xC2 < a,b | a4=b2=aba-1b > 8_3 dihedral group Dih8 (square) < a,c | a4=c2=acac > 8_3b dihedral group Dih8 (Heisenberg) < a,b,c | a2=b2=c2=abcbc > 8_4 dicyclic group Dic8, quaternion Q8 < a,d | a4=d4=adad-1=a2d2 > 8_4b dicyclic group Dic8, quaternion Q8 with i,j < i,j | i4=j4=i2j2=ijij-1 > 8_4c dicyclic group Dic8, quaternion Q8 with i,j,k < i,j,k | i4=j4=k4=i2j2=j2k2=kji > 8_5 Klein group K8 < a,b,c | a2=b2=(ab)2=c2=(bc)2=(ca)2 > 12_2a cyclic group C12 < a | a12 > 12_2c group C4xC3 < a,b | a4=b3=aba-1b-1 > 12_5a group C6xC2 < a,b | a6=b2=aba-1b > 12_5c group K4xC3 < a,b,c | a2=b2=(ab)2=c3=acac-1=bcbc-1 > 12_4a dihedral group Dih12 < a,c | a6=c2=acac > 12_4c group K4:C3 < a,b,c | a2=b2=(ab)2=c3=acac=bcbc-1 > 12_1a dicyclic group Dic12 < a,d | a6=d4=adad-1=a3d2 > 12_3a alternating group A4 (tetrahedron) < a,c | a3=c2=(ac)3 > 12_3b alternating group A4 (tetrahedron) < s,t | s3=t3=(st)2 > 12_3c group K4:C3 (A4) < a,b,c | a2=b2=(ab)2=c3=acabc-1=abcbc-1 > 16_1 cyclic group C16 < a | a16 > 16_5 group C8xC2 < a,b | a8=b2=aba-1b > 16_2 group C4xC4 < a,b | a4=b4=aba-1b-1 > 16_4 group C4:C4 < a,b | a4=b4=abab-1 > 16_10 group (C4xC2)xC2 < a,b,c | a4=b2=c2=aba-1b=aca-1c=bcbc > 16_3 group K8:C2 < a,b | a4=b2=(ab)4=(aab)2 > 16_12 group Q8xC2 < a,d,b | a4=d4=b2=a2dd=adad-1=aba-1b=dbd-1b > 16_11 group D8xC2 < a,b,c | a4=b2=c2=aba-1b=acac=abcabc > 16_13 group Cb8:C2 < a,b,c | a4=b2=c2=aba-1b=acac=a2bcbc > 16_13p Pauli group G1 < a,b,c | a2=b2=c2=abcacb=abacbc ... (ab)4=(bc)4=(ca)4=(abc)4 > 16_7 dihedral group Dih16 < a,c | a8=c2=acac > 16_8 quasidihedral group QD16 < a,c | a8=c2=aca-3c > 16_6 modular group M16 < a,b | a8=b2=aba3b > 16_9 dicyclic group Q16 < a,d | a8=d4=adad-1=a4d2 > 16_14 Klein group K16 < a,b,c,d | a2=b2=(ab)2=c2=(bc)2=(ca)2=d2=(ad)2=(bd)2=(cd)2 > 20_3 Frobenius group F20 < a,b | a5=b4=(ab)4=aaba-1b-1 > 21_1 Frobenius group F21 < a,b | a7=b3=aba5b2 > 24_2a cyclic group C24 < a | a24 > 24_2b group C8xC3, Cc24 < a,b | a8=b3=aba-1b-1 > 24_1c group C12:C2 < a,c | a12=c8=a3c2=caca-2 > 24_1b group C6:C4 < a,b,c | a6=b12=ab-2=c8=acac-1=cb3c > 24_1a group C8:C3 < a,b | a8=b3=aba-1b > 24_10c modular group M24 < a,b | a12=b2=aba5b > 24_10b group Cb12:C2 < a,b,c | a6=b2=aba-1b=c2=aca-1c=a3bcbc > 24_10a group Dih8xC3 < a,c,b | a4=c2=acac=b3=aba-1b-1=bcb-1c > 24_6c dihedral group Dih24 < a,c | a12=c2=acac > 24_6a group Dih8:C3 < a,c,b | a4=c2=acac=b3=aba-1b-1=bcbc > 24_9c group Cb24 < a,b | a12=b2=aba-1b > 24_9b group C6xC4 < a,b | a6=b4=aba-1b-1 > 24_9a group Cb8xC3 < a,b,c | a4=b2=aba-1b=c3=aca-1c-1=bcbc-1 > 24_5c quasidihedral group QD24 < a,c | a12=c2=aca-5c > 24_5b group Dih12:C2 < a,c,d | a6=c2=acac=d4=a3d2=acdcd > 24_5a group Cb8:C3 < a,b,c | a4=b2=aba-1b=c3=aca-1c-1=bcbc > 24_5f group Dih6xC4 < a,b,c | a3=b2=abab=c4=aca-1c-1=bcb-1c-1 > 24_11c group C12:C2 < a,b | a12=b12=a2b-2=a3bab > 24_11b group C6:C4 < a,b,c | a6=b12=ab-2=c12=b2c-2=abcbac-1 > 24_11a group Q8xC3 < a,d,b | a4=d4=adad-1=aadd=b3=aba-1b-1=dbd-1b-1 > 24_4c dicyclic group Dic24 < a,d | a12=d4=adad-1=a6d2 > 24_4b group (C4xC3):C2, Cc12:C2 < a,b,d | a4=b3=aba-1b=d4=adad-1=a2dd=dbd-1b > 24_4a group Q8:C3 < a,d,b | a4=d4=adad-1=a2dd=b3=aba-1b=dbd-1b > 24_7b group C6:C4 < a,b | a6=b4=abab-1 > 24_7d group Dic12xC2 < a,d,b | a6=d4=adad-1=a3dd=b2=aba-1b=dbd-1b > 24_7a group Cb8:C3 < a,b,c | a4=b2=aba-1b=c3=aca-1c=bcbc-1 > 24_7e group (K4xC3):C2 < a,b,c,d | a2=b2=c3=abab=acac-1=bcbc-1=d4=add=abdbd=acdcd > 24_8b group Cb12:C2 < a,b,c | a6=b2=aba-1b=c2=acac=a3bcbc > 24_8p group G2 < a,b,c | a2=b2=c2=abcbac=ababacbc=(ac)4=(bc)4=(ab)6 > 24_8d group Dic12:C2 < a,d,c | a6=d4=adad-1=a3dd=c2=acac=dcdc > 24_8a group Dih8:C3 < a,c,b | a4=c2=acac=b3=aba-1b=cbcb-1 > 24_8e group (K4xC3):C2 < a,b,c,d | a2=b2=c3=abab=acac-1=bcbc-1=d4=add=bdbd=acdcd > 24_15b group Cb12xC2 < a,b,c | a6=b2=aba-1b=c2=aca-1c=bcbc > 24_15a group K8xC3 < a,b,c | a6=b2=aba-1b=c2=aca-1c=bcbc > 24_15e group (K4xC3)xC2 < a,b,c,d | a2=b2=c3=abab=acac-1=bcbc-1=d2=adad=bdbd=cdc-1d > 24_14b group Dih12xC2 < a,b,c | a6=c2=acac=b2=aba-1b=bcbc > 24_14a group K8:C3 < a,b,c | a6=b2=aba-1b=c2=acac=bcbc > 24_14e group (K4xC3):C2 < a,b,c,d | a2=b2=c3=abab=acac-1=bcbc-1=d2=adad=bdbd=cdcd > 24_13a group A4xC2 < a,b | a3=b2=(aba-1b)2 > 24_12a symmetric group S4 (cube) < a,b | a3=b2=(abab)2 > 24_12p symmetric group S4 (cube) < a,b,c | a2=b2=c2=(ab)3=(ac)3=(bc)3=(babc)2=(cbca)2=(acab)2 > 24_3y group Q8:C3, special linear group SL(2,3), binary tetrahedral group < a,i | a6=i4=(ai)3=a3i2, ~e=a3=i2 > 24_3e group Q8:C3, special linear group SL(2,3) < a,i | a6=i4=(ai)6=a3i2=(ai)3i2, ~e=a3=i2=(ai)3 > 24_3f group Q8:C3, special linear group SL(2,3) < b,i | b3=i4=(bi)6=(bi)3i2, ~e=i2=(bi)3 > 24_3g group Q8:C3, special linear group SL(2,3) < a,b | a6=b3=(ab)4=(ab)2a3, ~e=a3=(ab)2 > 24_3c group Q8:C3, special linear group SL(2,3) < a,b | a3=b3=abab-1a-1b-1=(aab)4=(ab)6, ~e=(ab)3=(aab)2 > 24_3d group Q8:C3, special linear group SL(2,3) < a,b | a6=b6=a3b3=a3(ab)2=(ab)4, ~e=a3=b3=(ab)2 > 24_3a group Q8:C3, special linear group SL(2,3) < i,l | i4=l6=(il)3=i2l3 > 24_3x group Q8:C3, special linear group SL(2,3) < k,r | k6=r4=(kr)3=k3r2 > 24_3b group Q8:C3, special linear group SL(2,3) < i,j | i6=j4=(ij)3=i3j2 > 30_4a cyclic group C30 < a | a30 > 30_4b group C15xC2 < a,b | a15=b2=aba-1b-1 > 30_4c group C10xC3 < a,b | a10=b3=aba-1b-1 > 30_4d group C6xC5 < a,b | a6=b5=aba-1b-1 > 40_3a group C5:C8 < a,b | a5=b8=1, bab-1=a3 > 48_29x general linear group GL(2,3) < k,r | r2=k3=(kkrkrkr)2 ... (kr)8 > 60_5 alternating group A5 (icosahedron) < a,b | a2=b3=(ab)5 > 120_34 symmetric group S5 < a,b | a2=b4=(ab)5=(abab-1)3 > 120_5a SL(2,5) 168_42 projective special linear group PSL(2,7) < a,b | a2=b3=(ab)7=(abab-1)4 >