Orders with >300 groups of order n
n = 128, 192, 256, 288, 320, 384, 432, 448, 480.
[non]abelian, [non]soluble, supersoluble, [non]monomial, Z-groups, A-groups, metacyclic, metabelian, p-groups, elementary, hyperelementary, linear, perfect, simple, almost simple, quasisimple, rational groups.
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C1 | Trivial group | 1 | 1+ | C1 | 1,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C2 | Cyclic group | 2 | 1+ | C2 | 2,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C3 | Cyclic group; = A3 = triangle rotations | 3 | 1 | C3 | 3,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C4 | Cyclic group; = square rotations | 4 | 1 | C4 | 4,1 |
C22 | Klein 4-group V4 = elementary abelian group of type [2,2]; = rectangle symmetries | 4 | C2^2 | 4,2 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C5 | Cyclic group; = pentagon rotations | 5 | 1 | C5 | 5,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C6 | Cyclic group; = hexagon rotations | 6 | 1 | C6 | 6,2 |
S3 | Symmetric group on 3 letters; = D3 = GL2(F2) = triangle symmetries = 1st non-abelian group | 3 | 2+ | S3 | 6,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C7 | Cyclic group | 7 | 1 | C7 | 7,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C8 | Cyclic group | 8 | 1 | C8 | 8,1 |
D4 | Dihedral group; = He2 = AΣL1(F4) = 2+ 1+2 = square symmetries | 4 | 2+ | D4 | 8,3 |
Q8 | Quaternion group; = C4.C2 = Dic2 = 2- 1+2 | 8 | 2- | Q8 | 8,4 |
C23 | Elementary abelian group of type [2,2,2] | 8 | C2^3 | 8,5 | |
C2xC4 | Abelian group of type [2,4] | 8 | C2xC4 | 8,2 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C9 | Cyclic group | 9 | 1 | C9 | 9,1 |
C32 | Elementary abelian group of type [3,3] | 9 | C3^2 | 9,2 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C10 | Cyclic group | 10 | 1 | C10 | 10,2 |
D5 | Dihedral group; = pentagon symmetries | 5 | 2+ | D5 | 10,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C11 | Cyclic group | 11 | 1 | C11 | 11,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C12 | Cyclic group | 12 | 1 | C12 | 12,2 |
A4 | Alternating group on 4 letters; = PSL2(F3) = L2(3) = tetrahedron rotations | 4 | 3+ | A4 | 12,3 |
D6 | Dihedral group; = C2xS3 = hexagon symmetries | 6 | 2+ | D6 | 12,4 |
Dic3 | Dicyclic group; = C3:C4 | 12 | 2- | Dic3 | 12,1 |
C2xC6 | Abelian group of type [2,6] | 12 | C2xC6 | 12,5 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C13 | Cyclic group | 13 | 1 | C13 | 13,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C14 | Cyclic group | 14 | 1 | C14 | 14,2 |
D7 | Dihedral group | 7 | 2+ | D7 | 14,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C15 | Cyclic group | 15 | 1 | C15 | 15,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C16 | Cyclic group | 16 | 1 | C16 | 16,1 |
D8 | Dihedral group | 8 | 2+ | D8 | 16,7 |
Q16 | Generalised quaternion group; = C8.C2 = Dic4 | 16 | 2- | Q16 | 16,9 |
SD16 | Semidihedral group; = Q8:C2 = QD16 | 8 | 2 | SD16 | 16,8 |
M4(2) | Modular maximal-cyclic group; = C8:3C2 | 8 | 2 | M4(2) | 16,6 |
C4oD4 | Pauli group = central product of C4 and D4 | 8 | 2 | C4oD4 | 16,13 |
C22:C4 | The semidirect product of C22 and C4 acting via C4/C2=C2 | 8 | C2^2:C4 | 16,3 | |
C4:C4 | The semidirect product of C4 and C4 acting via C4/C2=C2 | 16 | C4:C4 | 16,4 | |
C42 | Abelian group of type [4,4] | 16 | C4^2 | 16,2 | |
C24 | Elementary abelian group of type [2,2,2,2] | 16 | C2^4 | 16,14 | |
C2xC8 | Abelian group of type [2,8] | 16 | C2xC8 | 16,5 | |
C22xC4 | Abelian group of type [2,2,4] | 16 | C2^2xC4 | 16,10 | |
C2xD4 | Direct product of C2 and D4 | 8 | C2xD4 | 16,11 | |
C2xQ8 | Direct product of C2 and Q8 | 16 | C2xQ8 | 16,12 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C17 | Cyclic group | 17 | 1 | C17 | 17,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C18 | Cyclic group | 18 | 1 | C18 | 18,2 |
D9 | Dihedral group | 9 | 2+ | D9 | 18,1 |
C3:S3 | The semidirect product of C3 and S3 acting via S3/C3=C2 | 9 | C3:S3 | 18,4 | |
C3xC6 | Abelian group of type [3,6] | 18 | C3xC6 | 18,5 | |
C3xS3 | Direct product of C3 and S3; = U2(F2) | 6 | 2 | C3xS3 | 18,3 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C19 | Cyclic group | 19 | 1 | C19 | 19,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C20 | Cyclic group | 20 | 1 | C20 | 20,2 |
D10 | Dihedral group; = C2xD5 | 10 | 2+ | D10 | 20,4 |
F5 | Frobenius group; = C5:C4 = AGL1(F5) = Aut(D5) = Hol(C5) = Sz(2) | 5 | 4+ | F5 | 20,3 |
Dic5 | Dicyclic group; = C5:2C4 | 20 | 2- | Dic5 | 20,1 |
C2xC10 | Abelian group of type [2,10] | 20 | C2xC10 | 20,5 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C21 | Cyclic group | 21 | 1 | C21 | 21,2 |
C7:C3 | The semidirect product of C7 and C3 acting faithfully | 7 | 3 | C7:C3 | 21,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C22 | Cyclic group | 22 | 1 | C22 | 22,2 |
D11 | Dihedral group | 11 | 2+ | D11 | 22,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C23 | Cyclic group | 23 | 1 | C23 | 23,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C24 | Cyclic group | 24 | 1 | C24 | 24,2 |
S4 | Symmetric group on 4 letters; = PGL2(F3) = Aut(Q8) = Hol(C22) = tetrahedron symmetries = cube/octahedron rotations | 4 | 3+ | S4 | 24,12 |
D12 | Dihedral group | 12 | 2+ | D12 | 24,6 |
Dic6 | Dicyclic group; = C3:Q8 | 24 | 2- | Dic6 | 24,4 |
SL2(F3) | Special linear group on F32; = Q8:C3 = 2T = <2,3,3> = 1st non-monomial group | 8 | 2- | SL(2,3) | 24,3 |
C3:D4 | The semidirect product of C3 and D4 acting via D4/C22=C2 | 12 | 2 | C3:D4 | 24,8 |
C3:C8 | The semidirect product of C3 and C8 acting via C8/C4=C2 | 24 | 2 | C3:C8 | 24,1 |
C2xC12 | Abelian group of type [2,12] | 24 | C2xC12 | 24,9 | |
C22xC6 | Abelian group of type [2,2,6] | 24 | C2^2xC6 | 24,15 | |
C2xA4 | Direct product of C2 and A4; = AΣL1(F8) | 6 | 3+ | C2xA4 | 24,13 |
C4xS3 | Direct product of C4 and S3 | 12 | 2 | C4xS3 | 24,5 |
C3xD4 | Direct product of C3 and D4 | 12 | 2 | C3xD4 | 24,10 |
C22xS3 | Direct product of C22 and S3 | 12 | C2^2xS3 | 24,14 | |
C3xQ8 | Direct product of C3 and Q8 | 24 | 2 | C3xQ8 | 24,11 |
C2xDic3 | Direct product of C2 and Dic3 | 24 | C2xDic3 | 24,7 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C25 | Cyclic group | 25 | 1 | C25 | 25,1 |
C52 | Elementary abelian group of type [5,5] | 25 | C5^2 | 25,2 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C26 | Cyclic group | 26 | 1 | C26 | 26,2 |
D13 | Dihedral group | 13 | 2+ | D13 | 26,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C27 | Cyclic group | 27 | 1 | C27 | 27,1 |
He3 | Heisenberg group; = C32:C3 = 3+ 1+2 | 9 | 3 | He3 | 27,3 |
3- 1+2 | Extraspecial group | 9 | 3 | ES-(3,1) | 27,4 |
C33 | Elementary abelian group of type [3,3,3] | 27 | C3^3 | 27,5 | |
C3xC9 | Abelian group of type [3,9] | 27 | C3xC9 | 27,2 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C28 | Cyclic group | 28 | 1 | C28 | 28,2 |
D14 | Dihedral group; = C2xD7 | 14 | 2+ | D14 | 28,3 |
Dic7 | Dicyclic group; = C7:C4 | 28 | 2- | Dic7 | 28,1 |
C2xC14 | Abelian group of type [2,14] | 28 | C2xC14 | 28,4 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C29 | Cyclic group | 29 | 1 | C29 | 29,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C30 | Cyclic group | 30 | 1 | C30 | 30,4 |
D15 | Dihedral group | 15 | 2+ | D15 | 30,3 |
C5xS3 | Direct product of C5 and S3 | 15 | 2 | C5xS3 | 30,1 |
C3xD5 | Direct product of C3 and D5 | 15 | 2 | C3xD5 | 30,2 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C31 | Cyclic group | 31 | 1 | C31 | 31,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C32 | Cyclic group | 32 | 1 | C32 | 32,1 |
D16 | Dihedral group | 16 | 2+ | D16 | 32,18 |
Q32 | Generalised quaternion group; = C16.C2 = Dic8 | 32 | 2- | Q32 | 32,20 |
2+ 1+4 | Extraspecial group; = D4oD4 | 8 | 4+ | ES+(2,2) | 32,49 |
SD32 | Semidihedral group; = C16:2C2 = QD32 | 16 | 2 | SD32 | 32,19 |
2- 1+4 | Gamma matrices = Extraspecial group; = D4oQ8 | 16 | 4- | ES-(2,2) | 32,50 |
M5(2) | Modular maximal-cyclic group; = C16:3C2 | 16 | 2 | M5(2) | 32,17 |
C4wrC2 | Wreath product of C4 by C2 | 8 | 2 | C4wrC2 | 32,11 |
C22wrC2 | Wreath product of C22 by C2 | 8 | C2^2wrC2 | 32,27 | |
C8oD4 | Central product of C8 and D4 | 16 | 2 | C8oD4 | 32,38 |
C4oD8 | Central product of C4 and D8 | 16 | 2 | C4oD8 | 32,42 |
C23:C4 | The semidirect product of C23 and C4 acting faithfully | 8 | 4+ | C2^3:C4 | 32,6 |
C8:C22 | The semidirect product of C8 and C22 acting faithfully; = Aut(D8) = Hol(C8) | 8 | 4+ | C8:C2^2 | 32,43 |
C4:D4 | The semidirect product of C4 and D4 acting via D4/C22=C2 | 16 | C4:D4 | 32,28 | |
C4:1D4 | The semidirect product of C4 and D4 acting via D4/C4=C2 | 16 | C4:1D4 | 32,34 | |
C22:C8 | The semidirect product of C22 and C8 acting via C8/C4=C2 | 16 | C2^2:C8 | 32,5 | |
C22:Q8 | The semidirect product of C22 and Q8 acting via Q8/C4=C2 | 16 | C2^2:Q8 | 32,29 | |
D4:C4 | 1st semidirect product of D4 and C4 acting via C4/C2=C2 | 16 | D4:C4 | 32,9 | |
C42:C2 | 1st semidirect product of C42 and C2 acting faithfully | 16 | C4^2:C2 | 32,24 | |
C42:2C2 | 2nd semidirect product of C42 and C2 acting faithfully | 16 | C4^2:2C2 | 32,33 | |
C4:C8 | The semidirect product of C4 and C8 acting via C8/C4=C2 | 32 | C4:C8 | 32,12 | |
C4:Q8 | The semidirect product of C4 and Q8 acting via Q8/C4=C2 | 32 | C4:Q8 | 32,35 | |
C8:C4 | 3rd semidirect product of C8 and C4 acting via C4/C2=C2 | 32 | C8:C4 | 32,4 | |
Q8:C4 | 1st semidirect product of Q8 and C4 acting via C4/C2=C2 | 32 | Q8:C4 | 32,10 | |
C4.D4 | 1st non-split extension by C4 of D4 acting via D4/C22=C2 | 8 | 4+ | C4.D4 | 32,7 |
C8.C4 | 1st non-split extension by C8 of C4 acting via C4/C2=C2 | 16 | 2 | C8.C4 | 32,15 |
C4.4D4 | 4th non-split extension by C4 of D4 acting via D4/C4=C2 | 16 | C4.4D4 | 32,31 | |
C8.C22 | The non-split extension by C8 of C22 acting faithfully | 16 | 4- | C8.C2^2 | 32,44 |
C4.10D4 | 2nd non-split extension by C4 of D4 acting via D4/C22=C2 | 16 | 4- | C4.10D4 | 32,8 |
C22.D4 | 3rd non-split extension by C22 of D4 acting via D4/C22=C2 | 16 | C2^2.D4 | 32,30 | |
C2.D8 | 2nd central extension by C2 of D8 | 32 | C2.D8 | 32,14 | |
C4.Q8 | 1st non-split extension by C4 of Q8 acting via Q8/C4=C2 | 32 | C4.Q8 | 32,13 | |
C2.C42 | 1st central stem extension by C2 of C42 | 32 | C2.C4^2 | 32,2 | |
C42.C2 | 4th non-split extension by C42 of C2 acting faithfully | 32 | C4^2.C2 | 32,32 | |
C25 | Elementary abelian group of type [2,2,2,2,2] | 32 | C2^5 | 32,51 | |
C4xC8 | Abelian group of type [4,8] | 32 | C4xC8 | 32,3 | |
C2xC16 | Abelian group of type [2,16] | 32 | C2xC16 | 32,16 | |
C2xC42 | Abelian group of type [2,4,4] | 32 | C2xC4^2 | 32,21 | |
C22xC8 | Abelian group of type [2,2,8] | 32 | C2^2xC8 | 32,36 | |
C23xC4 | Abelian group of type [2,2,2,4] | 32 | C2^3xC4 | 32,45 | |
C4xD4 | Direct product of C4 and D4 | 16 | C4xD4 | 32,25 | |
C2xD8 | Direct product of C2 and D8 | 16 | C2xD8 | 32,39 | |
C2xSD16 | Direct product of C2 and SD16 | 16 | C2xSD16 | 32,40 | |
C22xD4 | Direct product of C22 and D4 | 16 | C2^2xD4 | 32,46 | |
C2xM4(2) | Direct product of C2 and M4(2) | 16 | C2xM4(2) | 32,37 | |
C4xQ8 | Direct product of C4 and Q8 | 32 | C4xQ8 | 32,26 | |
C2xQ16 | Direct product of C2 and Q16 | 32 | C2xQ16 | 32,41 | |
C22xQ8 | Direct product of C22 and Q8 | 32 | C2^2xQ8 | 32,47 | |
C2xC4oD4 | Direct product of C2 and C4oD4 | 16 | C2xC4oD4 | 32,48 | |
C2xC22:C4 | Direct product of C2 and C22:C4 | 16 | C2xC2^2:C4 | 32,22 | |
C2xC4:C4 | Direct product of C2 and C4:C4 | 32 | C2xC4:C4 | 32,23 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C33 | Cyclic group | 33 | 1 | C33 | 33,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C34 | Cyclic group | 34 | 1 | C34 | 34,2 |
D17 | Dihedral group | 17 | 2+ | D17 | 34,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C35 | Cyclic group | 35 | 1 | C35 | 35,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C36 | Cyclic group | 36 | 1 | C36 | 36,2 |
D18 | Dihedral group; = C2xD9 | 18 | 2+ | D18 | 36,4 |
Dic9 | Dicyclic group; = C9:C4 | 36 | 2- | Dic9 | 36,1 |
C32:C4 | The semidirect product of C32 and C4 acting faithfully | 6 | 4+ | C3^2:C4 | 36,9 |
C3:Dic3 | The semidirect product of C3 and Dic3 acting via Dic3/C6=C2 | 36 | C3:Dic3 | 36,7 | |
C3.A4 | The central extension by C3 of A4 | 18 | 3 | C3.A4 | 36,3 |
C62 | Abelian group of type [6,6] | 36 | C6^2 | 36,14 | |
C2xC18 | Abelian group of type [2,18] | 36 | C2xC18 | 36,5 | |
C3xC12 | Abelian group of type [3,12] | 36 | C3xC12 | 36,8 | |
S32 | Direct product of S3 and S3; = Spin+4(F2) = Hol(S3) | 6 | 4+ | S3^2 | 36,10 |
S3xC6 | Direct product of C6 and S3 | 12 | 2 | S3xC6 | 36,12 |
C3xA4 | Direct product of C3 and A4 | 12 | 3 | C3xA4 | 36,11 |
C3xDic3 | Direct product of C3 and Dic3 | 12 | 2 | C3xDic3 | 36,6 |
C2xC3:S3 | Direct product of C2 and C3:S3 | 18 | C2xC3:S3 | 36,13 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C37 | Cyclic group | 37 | 1 | C37 | 37,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C38 | Cyclic group | 38 | 1 | C38 | 38,2 |
D19 | Dihedral group | 19 | 2+ | D19 | 38,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C39 | Cyclic group | 39 | 1 | C39 | 39,2 |
C13:C3 | The semidirect product of C13 and C3 acting faithfully | 13 | 3 | C13:C3 | 39,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C40 | Cyclic group | 40 | 1 | C40 | 40,2 |
D20 | Dihedral group | 20 | 2+ | D20 | 40,6 |
Dic10 | Dicyclic group; = C5:Q8 | 40 | 2- | Dic10 | 40,4 |
C5:D4 | The semidirect product of C5 and D4 acting via D4/C22=C2 | 20 | 2 | C5:D4 | 40,8 |
C5:C8 | The semidirect product of C5 and C8 acting via C8/C2=C4 | 40 | 4- | C5:C8 | 40,3 |
C5:2C8 | The semidirect product of C5 and C8 acting via C8/C4=C2 | 40 | 2 | C5:2C8 | 40,1 |
C2xC20 | Abelian group of type [2,20] | 40 | C2xC20 | 40,9 | |
C22xC10 | Abelian group of type [2,2,10] | 40 | C2^2xC10 | 40,14 | |
C2xF5 | Direct product of C2 and F5; = Aut(D10) = Hol(C10) | 10 | 4+ | C2xF5 | 40,12 |
C4xD5 | Direct product of C4 and D5 | 20 | 2 | C4xD5 | 40,5 |
C5xD4 | Direct product of C5 and D4 | 20 | 2 | C5xD4 | 40,10 |
C22xD5 | Direct product of C22 and D5 | 20 | C2^2xD5 | 40,13 | |
C5xQ8 | Direct product of C5 and Q8 | 40 | 2 | C5xQ8 | 40,11 |
C2xDic5 | Direct product of C2 and Dic5 | 40 | C2xDic5 | 40,7 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C41 | Cyclic group | 41 | 1 | C41 | 41,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C42 | Cyclic group | 42 | 1 | C42 | 42,6 |
D21 | Dihedral group | 21 | 2+ | D21 | 42,5 |
F7 | Frobenius group; = C7:C6 = AGL1(F7) = Aut(D7) = Hol(C7) | 7 | 6+ | F7 | 42,1 |
S3xC7 | Direct product of C7 and S3 | 21 | 2 | S3xC7 | 42,3 |
C3xD7 | Direct product of C3 and D7 | 21 | 2 | C3xD7 | 42,4 |
C2xC7:C3 | Direct product of C2 and C7:C3 | 14 | 3 | C2xC7:C3 | 42,2 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C43 | Cyclic group | 43 | 1 | C43 | 43,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C44 | Cyclic group | 44 | 1 | C44 | 44,2 |
D22 | Dihedral group; = C2xD11 | 22 | 2+ | D22 | 44,3 |
Dic11 | Dicyclic group; = C11:C4 | 44 | 2- | Dic11 | 44,1 |
C2xC22 | Abelian group of type [2,22] | 44 | C2xC22 | 44,4 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C45 | Cyclic group | 45 | 1 | C45 | 45,1 |
C3xC15 | Abelian group of type [3,15] | 45 | C3xC15 | 45,2 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C46 | Cyclic group | 46 | 1 | C46 | 46,2 |
D23 | Dihedral group | 23 | 2+ | D23 | 46,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C47 | Cyclic group | 47 | 1 | C47 | 47,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C49 | Cyclic group | 49 | 1 | C49 | 49,1 |
C72 | Elementary abelian group of type [7,7] | 49 | C7^2 | 49,2 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C50 | Cyclic group | 50 | 1 | C50 | 50,2 |
D25 | Dihedral group | 25 | 2+ | D25 | 50,1 |
C5:D5 | The semidirect product of C5 and D5 acting via D5/C5=C2 | 25 | C5:D5 | 50,4 | |
C5xC10 | Abelian group of type [5,10] | 50 | C5xC10 | 50,5 | |
C5xD5 | Direct product of C5 and D5; = AΣL1(F25) | 10 | 2 | C5xD5 | 50,3 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C51 | Cyclic group | 51 | 1 | C51 | 51,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C52 | Cyclic group | 52 | 1 | C52 | 52,2 |
D26 | Dihedral group; = C2xD13 | 26 | 2+ | D26 | 52,4 |
Dic13 | Dicyclic group; = C13:2C4 | 52 | 2- | Dic13 | 52,1 |
C13:C4 | The semidirect product of C13 and C4 acting faithfully | 13 | 4+ | C13:C4 | 52,3 |
C2xC26 | Abelian group of type [2,26] | 52 | C2xC26 | 52,5 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C53 | Cyclic group | 53 | 1 | C53 | 53,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C54 | Cyclic group | 54 | 1 | C54 | 54,2 |
D27 | Dihedral group | 27 | 2+ | D27 | 54,1 |
C9:C6 | The semidirect product of C9 and C6 acting faithfully; = Aut(D9) = Hol(C9) | 9 | 6+ | C9:C6 | 54,6 |
C32:C6 | The semidirect product of C32 and C6 acting faithfully | 9 | 6+ | C3^2:C6 | 54,5 |
He3:C2 | 2nd semidirect product of He3 and C2 acting faithfully; = Aut(3- 1+2) | 9 | 3 | He3:C2 | 54,8 |
C9:S3 | The semidirect product of C9 and S3 acting via S3/C3=C2 | 27 | C9:S3 | 54,7 | |
C33:C2 | 3rd semidirect product of C33 and C2 acting faithfully | 27 | C3^3:C2 | 54,14 | |
C3xC18 | Abelian group of type [3,18] | 54 | C3xC18 | 54,9 | |
C32xC6 | Abelian group of type [3,3,6] | 54 | C3^2xC6 | 54,15 | |
S3xC9 | Direct product of C9 and S3 | 18 | 2 | S3xC9 | 54,4 |
C3xD9 | Direct product of C3 and D9 | 18 | 2 | C3xD9 | 54,3 |
C2xHe3 | Direct product of C2 and He3 | 18 | 3 | C2xHe3 | 54,10 |
S3xC32 | Direct product of C32 and S3 | 18 | S3xC3^2 | 54,12 | |
C2x3- 1+2 | Direct product of C2 and 3- 1+2 | 18 | 3 | C2xES-(3,1) | 54,11 |
C3xC3:S3 | Direct product of C3 and C3:S3 | 18 | C3xC3:S3 | 54,13 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C55 | Cyclic group | 55 | 1 | C55 | 55,2 |
C11:C5 | The semidirect product of C11 and C5 acting faithfully | 11 | 5 | C11:C5 | 55,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C56 | Cyclic group | 56 | 1 | C56 | 56,2 |
D28 | Dihedral group | 28 | 2+ | D28 | 56,5 |
F8 | Frobenius group; = C23:C7 = AGL1(F8) | 8 | 7+ | F8 | 56,11 |
Dic14 | Dicyclic group; = C7:Q8 | 56 | 2- | Dic14 | 56,3 |
C7:D4 | The semidirect product of C7 and D4 acting via D4/C22=C2 | 28 | 2 | C7:D4 | 56,7 |
C7:C8 | The semidirect product of C7 and C8 acting via C8/C4=C2 | 56 | 2 | C7:C8 | 56,1 |
C2xC28 | Abelian group of type [2,28] | 56 | C2xC28 | 56,8 | |
C22xC14 | Abelian group of type [2,2,14] | 56 | C2^2xC14 | 56,13 | |
C4xD7 | Direct product of C4 and D7 | 28 | 2 | C4xD7 | 56,4 |
C7xD4 | Direct product of C7 and D4 | 28 | 2 | C7xD4 | 56,9 |
C22xD7 | Direct product of C22 and D7 | 28 | C2^2xD7 | 56,12 | |
C7xQ8 | Direct product of C7 and Q8 | 56 | 2 | C7xQ8 | 56,10 |
C2xDic7 | Direct product of C2 and Dic7 | 56 | C2xDic7 | 56,6 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C57 | Cyclic group | 57 | 1 | C57 | 57,2 |
C19:C3 | The semidirect product of C19 and C3 acting faithfully | 19 | 3 | C19:C3 | 57,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C58 | Cyclic group | 58 | 1 | C58 | 58,2 |
D29 | Dihedral group | 29 | 2+ | D29 | 58,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C59 | Cyclic group | 59 | 1 | C59 | 59,1 |
d | ρ | Label | ID | ||
---|---|---|---|---|---|
C60 | Cyclic group | 60 | 1 | C60 | 60,4 |
A5 | Alternating group on 5 letters; = SL2(F4) = L2(5) = L2(4) = icosahedron/dodecahedron rotations; 1st non-abelian simple | 5 | 3+ | A5 | 60,5 |
D30 | Dihedral group; = C2xD15 | 30 | 2+ | D30 | 60,12 |
Dic15 | Dicyclic group; = C3:Dic5 | 60 | 2- | Dic15 | 60,3 |
C3:F5 | The semidirect product of C3 and F5 acting via F5/D5=C2 | 15 | 4 | C3:F5 | 60,7 |
C2xC30 | Abelian group of type [2,30] | 60 | C2xC30 | 60,13 | |
S3xD5 | Direct product of S3 and D5 | 15 | 4+ | S3xD5 | 60,8 |
C3xF5 | Direct product of C3 and F5 | 15 | 4 | C3xF5 | 60,6 |
C5xA4 | Direct product of C5 and A4 | 20 | 3 | C5xA4 | 60,9 |
C6xD5 | Direct product of C6 and D5 | 30 | 2 | C6xD5 | 60,10 |
S3xC10 | Direct product of C10 and S3 | 30 | 2 | S3xC10 | 60,11 |
C5xDic3 | Direct product of C5 and Dic3 | 60 | 2 | C5xDic3 | 60,1 |
C3xDic5 | Direct product of C3 and Dic5 | 60 | 2 | C3xDic5 | 60,2 |