When G acts on a (finite) set, the set is
a disjoint union of orbits, the *transitive* G-sets. There is a natural
bijection

{transitive G-sets up to ≅} | ↔ | {subgroups of G up to conjugacy} |

X | ↦ | stabiliser of a point |

G/H | ↤ | H |

Transitive G-sets on which G acts *faithfully* correspond
to subgroups H with *trivial core* (or *core-free*),
that is those where the intersection of H with all of its conjugates is
trivial; equivalently, H contains no non-trivial normal subgroup of G.
In this case G can be viewed as a transitive subgroup of S_{n}
for *n*=(G:H), the index of H in G, called the * transitive degree*.
Conversely, all transitive subgroups of
S_{n} arise in this way. The transitive group database in GAP and Magma
contains all transitive subgroups of S_{n} up to conjugacy for *n*≤31,
numbered *n*T*i* (or T_{n,i}).

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{1} | Trivial group | C1 | 1,1 | 1T1 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{2} | Cyclic group | C2 | 2,1 | 2T1 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{3} | Cyclic group; = A_{3} = triangle rotations | C3 | 3,1 | 3T1 |

Label | ID | Tr ID | ||
---|---|---|---|---|

S_{3} | Symmetric group on 3 letters; = D_{3} = GL_{2}(𝔽_{2}) = triangle symmetries = 1^{st} non-abelian group | S3 | 6,1 | 3T2 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{4} | Cyclic group; = square rotations | C4 | 4,1 | 4T1 |

C_{2}^{2} | Klein 4-group V_{4} = elementary abelian group of type [2,2]; = rectangle symmetries | C2^2 | 4,2 | 4T2 |

Label | ID | Tr ID | ||
---|---|---|---|---|

D_{4} | Dihedral group; = He_{2} = AΣL_{1}(𝔽_{4}) = 2_{+}^{ 1+2} = square symmetries | D4 | 8,3 | 4T3 |

Label | ID | Tr ID | ||
---|---|---|---|---|

A_{4} | Alternating group on 4 letters; = PSL_{2}(𝔽_{3}) = L_{2}(3) = tetrahedron rotations | A4 | 12,3 | 4T4 |

Label | ID | Tr ID | ||
---|---|---|---|---|

S_{4} | Symmetric group on 4 letters; = PGL_{2}(𝔽_{3}) = Aut(Q_{8}) = Hol(C_{2}^{2}) = tetrahedron symmetries = cube/octahedron rotations | S4 | 24,12 | 4T5 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{5} | Cyclic group; = pentagon rotations | C5 | 5,1 | 5T1 |

Label | ID | Tr ID | ||
---|---|---|---|---|

D_{5} | Dihedral group; = pentagon symmetries | D5 | 10,1 | 5T2 |

Label | ID | Tr ID | ||
---|---|---|---|---|

F_{5} | Frobenius group; = C_{5}⋊C_{4} = AGL_{1}(𝔽_{5}) = Aut(D_{5}) = Hol(C_{5}) = Sz(2) | F5 | 20,3 | 5T3 |

Label | ID | Tr ID | ||
---|---|---|---|---|

A_{5} | Alternating group on 5 letters; = SL_{2}(𝔽_{4}) = L_{2}(5) = L_{2}(4) = icosahedron/dodecahedron rotations; 1^{st} non-abelian simple | A5 | 60,5 | 5T4 |

Label | ID | Tr ID | ||
---|---|---|---|---|

S_{5} | Symmetric group on 5 letters; = PGL_{2}(𝔽_{5}) = Aut(A_{5}) = 5-cell symmetries; almost simple | S5 | 120,34 | 5T5 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{6} | Cyclic group; = hexagon rotations | C6 | 6,2 | 6T1 |

S_{3} | Symmetric group on 3 letters; = D_{3} = GL_{2}(𝔽_{2}) = triangle symmetries = 1^{st} non-abelian group | S3 | 6,1 | 6T2 |

Label | ID | Tr ID | ||
---|---|---|---|---|

D_{6} | Dihedral group; = C_{2}×S_{3} = hexagon symmetries | D6 | 12,4 | 6T3 |

A_{4} | Alternating group on 4 letters; = PSL_{2}(𝔽_{3}) = L_{2}(3) = tetrahedron rotations | A4 | 12,3 | 6T4 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{3}×S_{3} | Direct product of C_{3} and S_{3}; = U_{2}(𝔽_{2}) | C3xS3 | 18,3 | 6T5 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{2}×A_{4} | Direct product of C_{2} and A_{4}; = AΣL_{1}(𝔽_{8}) | C2xA4 | 24,13 | 6T6 |

S_{4} | Symmetric group on 4 letters; = PGL_{2}(𝔽_{3}) = Aut(Q_{8}) = Hol(C_{2}^{2}) = tetrahedron symmetries = cube/octahedron rotations | S4 | 24,12 | 6T7 |

6T8 |

Label | ID | Tr ID | ||
---|---|---|---|---|

S_{3}^{2} | Direct product of S_{3} and S_{3}; = Spin+_{4}(𝔽_{2}) = Hol(S_{3}) | S3^2 | 36,10 | 6T9 |

C_{3}^{2}⋊C_{4} | The semidirect product of C_{3}^{2} and C_{4} acting faithfully | C3^2:C4 | 36,9 | 6T10 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{2}×S_{4} | Direct product of C_{2} and S_{4}; = O_{3}(𝔽_{3}) = cube/octahedron symmetries | C2xS4 | 48,48 | 6T11 |

Label | ID | Tr ID | ||
---|---|---|---|---|

A_{5} | Alternating group on 5 letters; = SL_{2}(𝔽_{4}) = L_{2}(5) = L_{2}(4) = icosahedron/dodecahedron rotations; 1^{st} non-abelian simple | A5 | 60,5 | 6T12 |

Label | ID | Tr ID | ||
---|---|---|---|---|

S_{3}≀C_{2} | Wreath product of S_{3} by C_{2}; = SO+_{4}(𝔽_{2}) | S3wrC2 | 72,40 | 6T13 |

Label | ID | Tr ID | ||
---|---|---|---|---|

S_{5} | Symmetric group on 5 letters; = PGL_{2}(𝔽_{5}) = Aut(A_{5}) = 5-cell symmetries; almost simple | S5 | 120,34 | 6T14 |

Label | ID | Tr ID | ||
---|---|---|---|---|

A_{6} | Alternating group on 6 letters; = PSL_{2}(𝔽_{9}) = L_{2}(9); 3^{rd} non-abelian simple | A6 | 360,118 | 6T15 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{7} | Cyclic group | C7 | 7,1 | 7T1 |

Label | ID | Tr ID | ||
---|---|---|---|---|

D_{7} | Dihedral group | D7 | 14,1 | 7T2 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{7}⋊C_{3} | The semidirect product of C_{7} and C_{3} acting faithfully | C7:C3 | 21,1 | 7T3 |

Label | ID | Tr ID | ||
---|---|---|---|---|

F_{7} | Frobenius group; = C_{7}⋊C_{6} = AGL_{1}(𝔽_{7}) = Aut(D_{7}) = Hol(C_{7}) | F7 | 42,1 | 7T4 |

Label | ID | Tr ID | ||
---|---|---|---|---|

GL_{3}(𝔽_{2}) | General linear group on 𝔽_{2}^{3}; = Aut(C_{2}^{3}) = L_{3}(2) = L_{2}(7); 2^{nd} non-abelian simple | GL(3,2) | 168,42 | 7T5 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{8} | Cyclic group | C8 | 8,1 | 8T1 |

C_{2}×C_{4} | Abelian group of type [2,4] | C2xC4 | 8,2 | 8T2 |

C_{2}^{3} | Elementary abelian group of type [2,2,2] | C2^3 | 8,5 | 8T3 |

D_{4} | Dihedral group; = He_{2} = AΣL_{1}(𝔽_{4}) = 2_{+}^{ 1+2} = square symmetries | D4 | 8,3 | 8T4 |

Q_{8} | Quaternion group; = C_{4}.C_{2} = Dic_{2} = 2_{-}^{ 1+2} | Q8 | 8,4 | 8T5 |

Label | ID | Tr ID | ||
---|---|---|---|---|

D_{8} | Dihedral group | D8 | 16,7 | 8T6 |

M_{4}(2) | Modular maximal-cyclic group; = C_{8}⋊_{3}C_{2} | M4(2) | 16,6 | 8T7 |

SD_{16} | Semidihedral group; = Q_{8}⋊C_{2} = QD_{16} | SD16 | 16,8 | 8T8 |

C_{2}×D_{4} | Direct product of C_{2} and D_{4} | C2xD4 | 16,11 | 8T9 |

C_{2}^{2}⋊C_{4} | The semidirect product of C_{2}^{2} and C_{4} acting via C_{4}/C_{2}=C_{2} | C2^2:C4 | 16,3 | 8T10 |

C_{4}○D_{4} | Pauli group = central product of C_{4} and D_{4} | C4oD4 | 16,13 | 8T11 |

Label | ID | Tr ID | ||
---|---|---|---|---|

SL_{2}(𝔽_{3}) | Special linear group on 𝔽_{3}^{2}; = Q_{8}⋊C_{3} = 2T = <2,3,3> = 1^{st} non-monomial group | SL(2,3) | 24,3 | 8T12 |

C_{2}×A_{4} | Direct product of C_{2} and A_{4}; = AΣL_{1}(𝔽_{8}) | C2xA4 | 24,13 | 8T13 |

S_{4} | Symmetric group on 4 letters; = PGL_{2}(𝔽_{3}) = Aut(Q_{8}) = Hol(C_{2}^{2}) = tetrahedron symmetries = cube/octahedron rotations | S4 | 24,12 | 8T14 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{8}⋊C_{2}^{2} | The semidirect product of C_{8} and C_{2}^{2} acting faithfully; = Aut(D_{8}) = Hol(C_{8}) | C8:C2^2 | 32,43 | 8T15 |

C_{4}.D_{4} | 1^{st} non-split extension by C_{4} of D_{4} acting via D_{4}/C_{2}^{2}=C_{2} | C4.D4 | 32,7 | 8T16 |

C_{4}≀C_{2} | Wreath product of C_{4} by C_{2} | C4wrC2 | 32,11 | 8T17 |

C_{2}^{2}≀C_{2} | Wreath product of C_{2}^{2} by C_{2} | C2^2wrC2 | 32,27 | 8T18 |

C_{2}^{3}⋊C_{4} | The semidirect product of C_{2}^{3} and C_{4} acting faithfully | C2^3:C4 | 32,6 | 8T19 |

8T20 | ||||

8T21 | ||||

2_{+}^{ 1+4} | Extraspecial group; = D_{4}○D_{4} | ES+(2,2) | 32,49 | 8T22 |

Label | ID | Tr ID | ||
---|---|---|---|---|

GL_{2}(𝔽_{3}) | General linear group on 𝔽_{3}^{2}; = Q_{8}⋊S_{3} = Aut(C_{3}^{2}) = 2O = <2,3,4> | GL(2,3) | 48,29 | 8T23 |

C_{2}×S_{4} | Direct product of C_{2} and S_{4}; = O_{3}(𝔽_{3}) = cube/octahedron symmetries | C2xS4 | 48,48 | 8T24 |

Label | ID | Tr ID | ||
---|---|---|---|---|

F_{8} | Frobenius group; = C_{2}^{3}⋊C_{7} = AGL_{1}(𝔽_{8}) | F8 | 56,11 | 8T25 |

Label | ID | Tr ID | ||
---|---|---|---|---|

D_{4}⋊_{4}D_{4} | 3^{rd} semidirect product of D_{4} and D_{4} acting via D_{4}/C_{2}^{2}=C_{2}; = Hol(D_{4}) | D4:4D4 | 64,134 | 8T26 |

C_{2}≀C_{4} | Wreath product of C_{2} by C_{4}; = AΣL_{1}(𝔽_{16}) | C2wrC4 | 64,32 | 8T27 |

8T28 | ||||

C_{2}≀C_{2}^{2} | Wreath product of C_{2} by C_{2}^{2}; = Hol(C_{2}×C_{4}) | C2wrC2^2 | 64,138 | 8T29 |

C_{4}^{2}⋊C_{4} | 2^{nd} semidirect product of C_{4}^{2} and C_{4} acting faithfully | C4^2:C4 | 64,34 | 8T30 |

C_{2}≀C_{2}^{2} | Wreath product of C_{2} by C_{2}^{2}; = Hol(C_{2}×C_{4}) | C2wrC2^2 | 64,138 | 8T31 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{2}^{3}⋊A_{4} | 2^{nd} semidirect product of C_{2}^{3} and A_{4} acting faithfully | C2^3:A4 | 96,204 | 8T32 |

C_{2}^{4}⋊C_{6} | 1^{st} semidirect product of C_{2}^{4} and C_{6} acting faithfully | C2^4:C6 | 96,70 | 8T33 |

C_{2}^{2}⋊S_{4} | The semidirect product of C_{2}^{2} and S_{4} acting via S_{4}/C_{2}^{2}=S_{3} | C2^2:S4 | 96,227 | 8T34 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{2}≀D_{4} | Wreath product of C_{2} by D_{4} | C2wrD4 | 128,928 | 8T35 |

Label | ID | Tr ID | ||
---|---|---|---|---|

AΓL_{1}(𝔽_{8}) | Affine semilinear group on 𝔽_{8}^{1}; = F_{8}⋊C_{3} = Aut(F_{8}) | AGammaL(1,8) | 168,43 | 8T36 |

GL_{3}(𝔽_{2}) | General linear group on 𝔽_{2}^{3}; = Aut(C_{2}^{3}) = L_{3}(2) = L_{2}(7); 2^{nd} non-abelian simple | GL(3,2) | 168,42 | 8T37 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{2}≀A_{4} | Wreath product of C_{2} by A_{4} | C2wrA4 | 192,201 | 8T38 |

C_{2}^{3}⋊S_{4} | 2^{nd} semidirect product of C_{2}^{3} and S_{4} acting faithfully; = Aut(C_{2}^{2}×C_{4}) | C2^3:S4 | 192,1493 | 8T39 |

Q_{8}⋊_{2}S_{4} | 2^{nd} semidirect product of Q_{8} and S_{4} acting via S_{4}/C_{2}^{2}=S_{3}; = Hol(Q_{8}) | Q8:2S4 | 192,1494 | 8T40 |

C_{2}^{4}⋊D_{6} | 1^{st} semidirect product of C_{2}^{4} and D_{6} acting faithfully; = Aut(C_{2}×Q_{8}) | C2^4:D6 | 192,955 | 8T41 |

Label | ID | Tr ID | ||
---|---|---|---|---|

A_{4}≀C_{2} | Wreath product of A_{4} by C_{2} | A4wrC2 | 288,1025 | 8T42 |

Label | ID | Tr ID | ||
---|---|---|---|---|

PGL_{2}(𝔽_{7}) | Projective linear group on 𝔽_{7}^{2}; = GL_{3}(𝔽_{2})⋊C_{2} = Aut(GL_{3}(𝔽_{2})); almost simple | PGL(2,7) | 336,208 | 8T43 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{9} | Cyclic group | C9 | 9,1 | 9T1 |

C_{3}^{2} | Elementary abelian group of type [3,3] | C3^2 | 9,2 | 9T2 |

Label | ID | Tr ID | ||
---|---|---|---|---|

D_{9} | Dihedral group | D9 | 18,1 | 9T3 |

C_{3}×S_{3} | Direct product of C_{3} and S_{3}; = U_{2}(𝔽_{2}) | C3xS3 | 18,3 | 9T4 |

C_{3}⋊S_{3} | The semidirect product of C_{3} and S_{3} acting via S_{3}/C_{3}=C_{2} | C3:S3 | 18,4 | 9T5 |

Label | ID | Tr ID | ||
---|---|---|---|---|

3_{-}^{ 1+2} | Extraspecial group | ES-(3,1) | 27,4 | 9T6 |

He_{3} | Heisenberg group; = C_{3}^{2}⋊C_{3} = 3_{+}^{ 1+2} | He3 | 27,3 | 9T7 |

Label | ID | Tr ID | ||
---|---|---|---|---|

S_{3}^{2} | Direct product of S_{3} and S_{3}; = Spin+_{4}(𝔽_{2}) = Hol(S_{3}) | S3^2 | 36,10 | 9T8 |

C_{3}^{2}⋊C_{4} | The semidirect product of C_{3}^{2} and C_{4} acting faithfully | C3^2:C4 | 36,9 | 9T9 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{9}⋊C_{6} | The semidirect product of C_{9} and C_{6} acting faithfully; = Aut(D_{9}) = Hol(C_{9}) | C9:C6 | 54,6 | 9T10 |

C_{3}^{2}⋊C_{6} | The semidirect product of C_{3}^{2} and C_{6} acting faithfully | C3^2:C6 | 54,5 | 9T11 |

He_{3}⋊C_{2} | 2^{nd} semidirect product of He_{3} and C_{2} acting faithfully; = Aut(3_{-}^{ 1+2}) | He3:C2 | 54,8 | 9T12 |

C_{3}^{2}⋊C_{6} | The semidirect product of C_{3}^{2} and C_{6} acting faithfully | C3^2:C6 | 54,5 | 9T13 |

Label | ID | Tr ID | ||
---|---|---|---|---|

PSU_{3}(𝔽_{2}) | Projective special unitary group on 𝔽_{2}^{3}; = C_{3}^{2}⋊Q_{8} = M_{9} | PSU(3,2) | 72,41 | 9T14 |

F_{9} | Frobenius group; = C_{3}^{2}⋊C_{8} = AGL_{1}(𝔽_{9}) | F9 | 72,39 | 9T15 |

S_{3}≀C_{2} | Wreath product of S_{3} by C_{2}; = SO+_{4}(𝔽_{2}) | S3wrC2 | 72,40 | 9T16 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{3}≀C_{3} | Wreath product of C_{3} by C_{3}; = AΣL_{1}(𝔽_{27}) | C3wrC3 | 81,7 | 9T17 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{3}^{2}⋊D_{6} | The semidirect product of C_{3}^{2} and D_{6} acting faithfully | C3^2:D6 | 108,17 | 9T18 |

Label | ID | Tr ID | ||
---|---|---|---|---|

AΓL_{1}(𝔽_{9}) | Affine semilinear group on 𝔽_{9}^{1}; = F_{9}⋊C_{2} = Aut(C_{3}^{2}⋊C_{4}) | AGammaL(1,9) | 144,182 | 9T19 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{3}≀S_{3} | Wreath product of C_{3} by S_{3} | C3wrS3 | 162,10 | 9T20 |

C_{3}^{3}⋊S_{3} | 2^{nd} semidirect product of C_{3}^{3} and S_{3} acting faithfully | C3^3:S3 | 162,19 | 9T21 |

C_{3}^{3}⋊C_{6} | 1^{st} semidirect product of C_{3}^{3} and C_{6} acting faithfully | C3^3:C6 | 162,11 | 9T22 |

Label | ID | Tr ID | ||
---|---|---|---|---|

ASL_{2}(𝔽_{3}) | Hessian group = Affine special linear group on 𝔽_{3}^{2}; = PSU_{3}(𝔽_{2})⋊C_{3} | ASL(2,3) | 216,153 | 9T23 |

Label | ID | Tr ID | ||
---|---|---|---|---|

He_{3}⋊D_{6} | The semidirect product of He_{3} and D_{6} acting faithfully | He3:D6 | 324,39 | 9T24 |

C_{3}^{3}⋊A_{4} | The semidirect product of C_{3}^{3} and A_{4} acting faithfully | C3^3:A4 | 324,160 | 9T25 |

Label | ID | Tr ID | ||
---|---|---|---|---|

AGL_{2}(𝔽_{3}) | Affine linear group on 𝔽_{3}^{2}; = PSU_{3}(𝔽_{2})⋊S_{3} = Aut(C_{3}⋊S_{3}) = Hol(C_{3}^{2}) | AGL(2,3) | 432,734 | 9T26 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{10} | Cyclic group | C10 | 10,2 | 10T1 |

D_{5} | Dihedral group; = pentagon symmetries | D5 | 10,1 | 10T2 |

Label | ID | Tr ID | ||
---|---|---|---|---|

D_{10} | Dihedral group; = C_{2}×D_{5} | D10 | 20,4 | 10T3 |

F_{5} | Frobenius group; = C_{5}⋊C_{4} = AGL_{1}(𝔽_{5}) = Aut(D_{5}) = Hol(C_{5}) = Sz(2) | F5 | 20,3 | 10T4 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{2}×F_{5} | Direct product of C_{2} and F_{5}; = Aut(D_{10}) = Hol(C_{10}) | C2xF5 | 40,12 | 10T5 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{5}×D_{5} | Direct product of C_{5} and D_{5}; = AΣL_{1}(𝔽_{25}) | C5xD5 | 50,3 | 10T6 |

Label | ID | Tr ID | ||
---|---|---|---|---|

A_{5} | Alternating group on 5 letters; = SL_{2}(𝔽_{4}) = L_{2}(5) = L_{2}(4) = icosahedron/dodecahedron rotations; 1^{st} non-abelian simple | A5 | 60,5 | 10T7 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{2}^{4}⋊C_{5} | The semidirect product of C_{2}^{4} and C_{5} acting faithfully | C2^4:C5 | 80,49 | 10T8 |

Label | ID | Tr ID | ||
---|---|---|---|---|

D_{5}^{2} | Direct product of D_{5} and D_{5} | D5^2 | 100,13 | 10T9 |

C_{5}^{2}⋊C_{4} | 4^{th} semidirect product of C_{5}^{2} and C_{4} acting faithfully | C5^2:C4 | 100,12 | 10T10 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{2}×A_{5} | Direct product of C_{2} and A_{5}; = icosahedron/dodecahedron symmetries | C2xA5 | 120,35 | 10T11 |

S_{5} | Symmetric group on 5 letters; = PGL_{2}(𝔽_{5}) = Aut(A_{5}) = 5-cell symmetries; almost simple | S5 | 120,34 | 10T12 |

10T13 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{2}×C_{2}^{4}⋊C_{5} | Direct product of C_{2} and C_{2}^{4}⋊C_{5}; = AΣL_{1}(𝔽_{32}) | C2xC2^4:C5 | 160,235 | 10T14 |

C_{2}^{4}⋊D_{5} | The semidirect product of C_{2}^{4} and D_{5} acting faithfully | C2^4:D5 | 160,234 | 10T15 |

10T16 |

Label | ID | Tr ID | ||
---|---|---|---|---|

D_{5}⋊F_{5} | The semidirect product of D_{5} and F_{5} acting via F_{5}/D_{5}=C_{2}; = Hol(D_{5}) | D5:F5 | 200,42 | 10T17 |

C_{5}^{2}⋊C_{8} | The semidirect product of C_{5}^{2} and C_{8} acting faithfully | C5^2:C8 | 200,40 | 10T18 |

D_{5}≀C_{2} | Wreath product of D_{5} by C_{2} | D5wrC2 | 200,43 | 10T19 |

C_{5}^{2}⋊Q_{8} | The semidirect product of C_{5}^{2} and Q_{8} acting faithfully | C5^2:Q8 | 200,44 | 10T20 |

D_{5}≀C_{2} | Wreath product of D_{5} by C_{2} | D5wrC2 | 200,43 | 10T21 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{2}×S_{5} | Direct product of C_{2} and S_{5}; = O_{3}(𝔽_{5}) | C2xS5 | 240,189 | 10T22 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{2}×C_{2}^{4}⋊D_{5} | Direct product of C_{2} and C_{2}^{4}⋊D_{5} | C2xC2^4:D5 | 320,1636 | 10T23 |

C_{2}^{4}⋊F_{5} | The semidirect product of C_{2}^{4} and F_{5} acting faithfully | C2^4:F5 | 320,1635 | 10T24 |

10T25 |

Label | ID | Tr ID | ||
---|---|---|---|---|

A_{6} | Alternating group on 6 letters; = PSL_{2}(𝔽_{9}) = L_{2}(9); 3^{rd} non-abelian simple | A6 | 360,118 | 10T26 |

Label | ID | Tr ID | ||
---|---|---|---|---|

D_{5}≀C_{2}⋊C_{2} | The semidirect product of D_{5}≀C_{2} and C_{2} acting faithfully | D5wrC2:C2 | 400,207 | 10T27 |

C_{5}^{2}⋊M_{4}(2) | The semidirect product of C_{5}^{2} and M_{4}(2) acting faithfully | C5^2:M4(2) | 400,206 | 10T28 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{11} | Cyclic group | C11 | 11,1 | 11T1 |

Label | ID | Tr ID | ||
---|---|---|---|---|

D_{11} | Dihedral group | D11 | 22,1 | 11T2 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{11}⋊C_{5} | The semidirect product of C_{11} and C_{5} acting faithfully | C11:C5 | 55,1 | 11T3 |

Label | ID | Tr ID | ||
---|---|---|---|---|

F_{11} | Frobenius group; = C_{11}⋊C_{10} = AGL_{1}(𝔽_{11}) = Aut(D_{11}) = Hol(C_{11}) | F11 | 110,1 | 11T4 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{12} | Cyclic group | C12 | 12,2 | 12T1 |

C_{2}×C_{6} | Abelian group of type [2,6] | C2xC6 | 12,5 | 12T2 |

D_{6} | Dihedral group; = C_{2}×S_{3} = hexagon symmetries | D6 | 12,4 | 12T3 |

A_{4} | Alternating group on 4 letters; = PSL_{2}(𝔽_{3}) = L_{2}(3) = tetrahedron rotations | A4 | 12,3 | 12T4 |

Dic_{3} | Dicyclic group; = C_{3}⋊C_{4} | Dic3 | 12,1 | 12T5 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{2}×A_{4} | Direct product of C_{2} and A_{4}; = AΣL_{1}(𝔽_{8}) | C2xA4 | 24,13 | 12T6 |

12T7 | ||||

S_{4} | Symmetric group on 4 letters; = PGL_{2}(𝔽_{3}) = Aut(Q_{8}) = Hol(C_{2}^{2}) = tetrahedron symmetries = cube/octahedron rotations | S4 | 24,12 | 12T8 |

12T9 | ||||

C_{2}^{2}×S_{3} | Direct product of C_{2}^{2} and S_{3} | C2^2xS3 | 24,14 | 12T10 |

C_{4}×S_{3} | Direct product of C_{4} and S_{3} | C4xS3 | 24,5 | 12T11 |

D_{12} | Dihedral group | D12 | 24,6 | 12T12 |

C_{3}⋊D_{4} | The semidirect product of C_{3} and D_{4} acting via D_{4}/C_{2}^{2}=C_{2} | C3:D4 | 24,8 | 12T13 |

C_{3}×D_{4} | Direct product of C_{3} and D_{4} | C3xD4 | 24,10 | 12T14 |

C_{3}⋊D_{4} | The semidirect product of C_{3} and D_{4} acting via D_{4}/C_{2}^{2}=C_{2} | C3:D4 | 24,8 | 12T15 |

Label | ID | Tr ID | ||
---|---|---|---|---|

S_{3}^{2} | Direct product of S_{3} and S_{3}; = Spin+_{4}(𝔽_{2}) = Hol(S_{3}) | S3^2 | 36,10 | 12T16 |

C_{3}^{2}⋊C_{4} | The semidirect product of C_{3}^{2} and C_{4} acting faithfully | C3^2:C4 | 36,9 | 12T17 |

S_{3}×C_{6} | Direct product of C_{6} and S_{3} | S3xC6 | 36,12 | 12T18 |

C_{3}×Dic_{3} | Direct product of C_{3} and Dic_{3} | C3xDic3 | 36,6 | 12T19 |

C_{3}×A_{4} | Direct product of C_{3} and A_{4} | C3xA4 | 36,11 | 12T20 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{2}×S_{4} | Direct product of C_{2} and S_{4}; = O_{3}(𝔽_{3}) = cube/octahedron symmetries | C2xS4 | 48,48 | 12T21 |

12T22 | ||||

12T23 | ||||

12T24 | ||||

C_{2}^{2}×A_{4} | Direct product of C_{2}^{2} and A_{4} | C2^2xA4 | 48,49 | 12T25 |

12T26 | ||||

A_{4}⋊C_{4} | The semidirect product of A_{4} and C_{4} acting via C_{4}/C_{2}=C_{2}; = SL_{2}(ℤ/4ℤ) | A4:C4 | 48,30 | 12T27 |

S_{3}×D_{4} | Direct product of S_{3} and D_{4}; = Aut(D_{12}) = Hol(C_{12}) | S3xD4 | 48,38 | 12T28 |

C_{4}×A_{4} | Direct product of C_{4} and A_{4} | C4xA4 | 48,31 | 12T29 |

A_{4}⋊C_{4} | The semidirect product of A_{4} and C_{4} acting via C_{4}/C_{2}=C_{2}; = SL_{2}(ℤ/4ℤ) | A4:C4 | 48,30 | 12T30 |

C_{4}^{2}⋊C_{3} | The semidirect product of C_{4}^{2} and C_{3} acting faithfully | C4^2:C3 | 48,3 | 12T31 |

C_{2}^{2}⋊A_{4} | The semidirect product of C_{2}^{2} and A_{4} acting via A_{4}/C_{2}^{2}=C_{3} | C2^2:A4 | 48,50 | 12T32 |

Label | ID | Tr ID | ||
---|---|---|---|---|

A_{5} | Alternating group on 5 letters; = SL_{2}(𝔽_{4}) = L_{2}(5) = L_{2}(4) = icosahedron/dodecahedron rotations; 1^{st} non-abelian simple | A5 | 60,5 | 12T33 |

Label | ID | Tr ID | ||
---|---|---|---|---|

S_{3}≀C_{2} | Wreath product of S_{3} by C_{2}; = SO+_{4}(𝔽_{2}) | S3wrC2 | 72,40 | 12T34 |

12T35 | ||||

12T36 | ||||

C_{2}×S_{3}^{2} | Direct product of C_{2}, S_{3} and S_{3} | C2xS3^2 | 72,46 | 12T37 |

C_{3}⋊D_{12} | The semidirect product of C_{3} and D_{12} acting via D_{12}/D_{6}=C_{2} | C3:D12 | 72,23 | 12T38 |

C_{6}.D_{6} | 2^{nd} non-split extension by C_{6} of D_{6} acting via D_{6}/S_{3}=C_{2} | C6.D6 | 72,21 | 12T39 |

C_{2}×C_{3}^{2}⋊C_{4} | Direct product of C_{2} and C_{3}^{2}⋊C_{4} | C2xC3^2:C4 | 72,45 | 12T40 |

12T41 | ||||

C_{3}×C_{3}⋊D_{4} | Direct product of C_{3} and C_{3}⋊D_{4} | C3xC3:D4 | 72,30 | 12T42 |

S_{3}×A_{4} | Direct product of S_{3} and A_{4} | S3xA4 | 72,44 | 12T43 |

C_{3}⋊S_{4} | The semidirect product of C_{3} and S_{4} acting via S_{4}/A_{4}=C_{2} | C3:S4 | 72,43 | 12T44 |

C_{3}×S_{4} | Direct product of C_{3} and S_{4} | C3xS4 | 72,42 | 12T45 |

F_{9} | Frobenius group; = C_{3}^{2}⋊C_{8} = AGL_{1}(𝔽_{9}) | F9 | 72,39 | 12T46 |

PSU_{3}(𝔽_{2}) | Projective special unitary group on 𝔽_{2}^{3}; = C_{3}^{2}⋊Q_{8} = M_{9} | PSU(3,2) | 72,41 | 12T47 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{3}×S_{3}^{2} | Direct product of C_{3}, S_{3} and S_{3} | C3xS3^2 | 108,38 | 12T70 |

C_{3}^{2}⋊_{4}D_{6} | The semidirect product of C_{3}^{2} and D_{6} acting via D_{6}/C_{3}=C_{2}^{2} | C3^2:4D6 | 108,40 | 12T71 |

C_{3}^{3}⋊C_{4} | 2^{nd} semidirect product of C_{3}^{3} and C_{4} acting faithfully | C3^3:C4 | 108,37 | 12T72 |

C_{3}×C_{3}^{2}⋊C_{4} | Direct product of C_{3} and C_{3}^{2}⋊C_{4} | C3xC3^2:C4 | 108,36 | 12T73 |

Label | ID | Tr ID | ||
---|---|---|---|---|

S_{5} | Symmetric group on 5 letters; = PGL_{2}(𝔽_{5}) = Aut(A_{5}) = 5-cell symmetries; almost simple | S5 | 120,34 | 12T74 |

C_{2}×A_{5} | Direct product of C_{2} and A_{5}; = icosahedron/dodecahedron symmetries | C2xA5 | 120,35 | 12T75 |

12T76 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{2}×S_{3}≀C_{2} | Direct product of C_{2} and S_{3}≀C_{2} | C2xS3wrC2 | 144,186 | 12T77 |

12T78 | ||||

(S_{3}^{2})⋊C_{4} | The semidirect product of S_{3}^{2} and C_{4} acting via C_{4}/C_{2}=C_{2} | (S3^2):C4 | 144,115 | 12T79 |

12T80 | ||||

Dic_{3}⋊D_{6} | 2^{nd} semidirect product of Dic_{3} and D_{6} acting via D_{6}/S_{3}=C_{2}; = Hol(Dic_{3}) | Dic3:D6 | 144,154 | 12T81 |

C_{6}^{2}⋊C_{4} | 1^{st} semidirect product of C_{6}^{2} and C_{4} acting faithfully | C6^2:C4 | 144,136 | 12T82 |

S_{3}×S_{4} | Direct product of S_{3} and S_{4}; = Hol(C_{2}×C_{6}) | S3xS4 | 144,183 | 12T83 |

AΓL_{1}(𝔽_{9}) | Affine semilinear group on 𝔽_{9}^{1}; = F_{9}⋊C_{2} = Aut(C_{3}^{2}⋊C_{4}) | AGammaL(1,9) | 144,182 | 12T84 |

A_{4}^{2} | Direct product of A_{4} and A_{4}; = PΩ+_{4}(𝔽_{3}) | A4^2 | 144,184 | 12T85 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{3}^{3}⋊D_{4} | 2^{nd} semidirect product of C_{3}^{3} and D_{4} acting faithfully | C3^3:D4 | 216,158 | 12T116 |

S_{3}^{3} | Direct product of S_{3}, S_{3} and S_{3}; = Hol(C_{3}×S_{3}) | S3^3 | 216,162 | 12T117 |

C_{3}^{2}⋊_{2}D_{12} | The semidirect product of C_{3}^{2} and D_{12} acting via D_{12}/C_{3}=D_{4} | C3^2:2D12 | 216,159 | 12T118 |

S_{3}×C_{3}^{2}⋊C_{4} | Direct product of S_{3} and C_{3}^{2}⋊C_{4} | S3xC3^2:C4 | 216,156 | 12T119 |

C_{3}^{3}⋊D_{4} | 2^{nd} semidirect product of C_{3}^{3} and D_{4} acting faithfully | C3^3:D4 | 216,158 | 12T120 |

C_{3}×S_{3}≀C_{2} | Direct product of C_{3} and S_{3}≀C_{2} | C3xS3wrC2 | 216,157 | 12T121 |

ASL_{2}(𝔽_{3}) | Hessian group = Affine special linear group on 𝔽_{3}^{2}; = PSU_{3}(𝔽_{2})⋊C_{3} | ASL(2,3) | 216,153 | 12T122 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{2}×S_{5} | Direct product of C_{2} and S_{5}; = O_{3}(𝔽_{5}) | C2xS5 | 240,189 | 12T123 |

A_{5}⋊C_{4} | The semidirect product of A_{5} and C_{4} acting via C_{4}/C_{2}=C_{2} | A5:C4 | 240,91 | 12T124 |

Label | ID | Tr ID | ||
---|---|---|---|---|

D_{6}≀C_{2} | Wreath product of D_{6} by C_{2} | D6wrC2 | 288,889 | 12T125 |

A_{4}≀C_{2} | Wreath product of A_{4} by C_{2} | A4wrC2 | 288,1025 | 12T126 |

PSO_{4}^{+} (𝔽_{3}) | Projective special orthogonal group of + type on 𝔽_{3}^{4}; = A_{4}⋊S_{4} = Hol(A_{4}) | PSO+(4,3) | 288,1026 | 12T127 |

A_{4}≀C_{2} | Wreath product of A_{4} by C_{2} | A4wrC2 | 288,1025 | 12T128 |

12T129 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{3}×C_{3}^{2}⋊_{4}D_{6} | Direct product of C_{3} and C_{3}^{2}⋊_{4}D_{6} | C3xC3^2:4D6 | 324,167 | 12T130 |

C_{3}×C_{3}^{3}⋊C_{4} | Direct product of C_{3} and C_{3}^{3}⋊C_{4}; = AΣL_{1}(𝔽_{81}) | C3xC3^3:C4 | 324,162 | 12T131 |

C_{3}^{3}⋊A_{4} | The semidirect product of C_{3}^{3} and A_{4} acting faithfully | C3^3:A4 | 324,160 | 12T132 |

12T133 |

Label | ID | Tr ID | ||
---|---|---|---|---|

S_{3}×S_{3}≀C_{2} | Direct product of S_{3} and S_{3}≀C_{2} | S3xS3wrC2 | 432,741 | 12T156 |

AGL_{2}(𝔽_{3}) | Affine linear group on 𝔽_{3}^{2}; = PSU_{3}(𝔽_{2})⋊S_{3} = Aut(C_{3}⋊S_{3}) = Hol(C_{3}^{2}) | AGL(2,3) | 432,734 | 12T157 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{13} | Cyclic group | C13 | 13,1 | 13T1 |

Label | ID | Tr ID | ||
---|---|---|---|---|

D_{13} | Dihedral group | D13 | 26,1 | 13T2 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{13}⋊C_{3} | The semidirect product of C_{13} and C_{3} acting faithfully | C13:C3 | 39,1 | 13T3 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{13}⋊C_{4} | The semidirect product of C_{13} and C_{4} acting faithfully | C13:C4 | 52,3 | 13T4 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{13}⋊C_{6} | The semidirect product of C_{13} and C_{6} acting faithfully | C13:C6 | 78,1 | 13T5 |

Label | ID | Tr ID | ||
---|---|---|---|---|

F_{13} | Frobenius group; = C_{13}⋊C_{12} = AGL_{1}(𝔽_{13}) = Aut(D_{13}) = Hol(C_{13}) | F13 | 156,7 | 13T6 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{14} | Cyclic group | C14 | 14,2 | 14T1 |

D_{7} | Dihedral group | D7 | 14,1 | 14T2 |

Label | ID | Tr ID | ||
---|---|---|---|---|

D_{14} | Dihedral group; = C_{2}×D_{7} | D14 | 28,3 | 14T3 |

Label | ID | Tr ID | ||
---|---|---|---|---|

F_{7} | Frobenius group; = C_{7}⋊C_{6} = AGL_{1}(𝔽_{7}) = Aut(D_{7}) = Hol(C_{7}) | F7 | 42,1 | 14T4 |

C_{2}×C_{7}⋊C_{3} | Direct product of C_{2} and C_{7}⋊C_{3} | C2xC7:C3 | 42,2 | 14T5 |

Label | ID | Tr ID | ||
---|---|---|---|---|

F_{8} | Frobenius group; = C_{2}^{3}⋊C_{7} = AGL_{1}(𝔽_{8}) | F8 | 56,11 | 14T6 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{2}×F_{7} | Direct product of C_{2} and F_{7}; = Aut(D_{14}) = Hol(C_{14}) | C2xF7 | 84,7 | 14T7 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{7}×D_{7} | Direct product of C_{7} and D_{7}; = AΣL_{1}(𝔽_{49}) | C7xD7 | 98,3 | 14T8 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{2}×F_{8} | Direct product of C_{2} and F_{8} | C2xF8 | 112,41 | 14T9 |

Label | ID | Tr ID | ||
---|---|---|---|---|

GL_{3}(𝔽_{2}) | General linear group on 𝔽_{2}^{3}; = Aut(C_{2}^{3}) = L_{3}(2) = L_{2}(7); 2^{nd} non-abelian simple | GL(3,2) | 168,42 | 14T10 |

AΓL_{1}(𝔽_{8}) | Affine semilinear group on 𝔽_{8}^{1}; = F_{8}⋊C_{3} = Aut(F_{8}) | AGammaL(1,8) | 168,43 | 14T11 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{7}^{2}⋊C_{4} | The semidirect product of C_{7}^{2} and C_{4} acting faithfully | C7^2:C4 | 196,8 | 14T12 |

D_{7}^{2} | Direct product of D_{7} and D_{7} | D7^2 | 196,9 | 14T13 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{7}⋊_{4}F_{7} | 2^{nd} semidirect product of C_{7} and F_{7} acting via F_{7}/D_{7}=C_{3} | C7:4F7 | 294,12 | 14T14 |

C_{7}^{2}⋊S_{3} | The semidirect product of C_{7}^{2} and S_{3} acting faithfully | C7^2:S3 | 294,7 | 14T15 |

Label | ID | Tr ID | ||
---|---|---|---|---|

PGL_{2}(𝔽_{7}) | Projective linear group on 𝔽_{7}^{2}; = GL_{3}(𝔽_{2})⋊C_{2} = Aut(GL_{3}(𝔽_{2})); almost simple | PGL(2,7) | 336,208 | 14T16 |

C_{2}×GL_{3}(𝔽_{2}) | Direct product of C_{2} and GL_{3}(𝔽_{2}) | C2xGL(3,2) | 336,209 | 14T17 |

C_{2}×AΓL_{1}(𝔽_{8}) | Direct product of C_{2} and AΓL_{1}(𝔽_{8}) | C2xAGammaL(1,8) | 336,210 | 14T18 |

C_{2}×GL_{3}(𝔽_{2}) | Direct product of C_{2} and GL_{3}(𝔽_{2}) | C2xGL(3,2) | 336,209 | 14T19 |

Label | ID | Tr ID | ||
---|---|---|---|---|

D_{7}≀C_{2} | Wreath product of D_{7} by C_{2} | D7wrC2 | 392,37 | 14T20 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{2}^{3}⋊F_{8} | 2^{nd} semidirect product of C_{2}^{3} and F_{8} acting via F_{8}/C_{2}^{3}=C_{7} | C2^3:F8 | 448,1394 | 14T21 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{15} | Cyclic group | C15 | 15,1 | 15T1 |

Label | ID | Tr ID | ||
---|---|---|---|---|

D_{15} | Dihedral group | D15 | 30,3 | 15T2 |

C_{3}×D_{5} | Direct product of C_{3} and D_{5} | C3xD5 | 30,2 | 15T3 |

C_{5}×S_{3} | Direct product of C_{5} and S_{3} | C5xS3 | 30,1 | 15T4 |

Label | ID | Tr ID | ||
---|---|---|---|---|

A_{5} | Alternating group on 5 letters; = SL_{2}(𝔽_{4}) = L_{2}(5) = L_{2}(4) = icosahedron/dodecahedron rotations; 1^{st} non-abelian simple | A5 | 60,5 | 15T5 |

C_{3}⋊F_{5} | The semidirect product of C_{3} and F_{5} acting via F_{5}/D_{5}=C_{2} | C3:F5 | 60,7 | 15T6 |

S_{3}×D_{5} | Direct product of S_{3} and D_{5} | S3xD5 | 60,8 | 15T7 |

C_{3}×F_{5} | Direct product of C_{3} and F_{5} | C3xF5 | 60,6 | 15T8 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{5}^{2}⋊C_{3} | The semidirect product of C_{5}^{2} and C_{3} acting faithfully | C5^2:C3 | 75,2 | 15T9 |

Label | ID | Tr ID | ||
---|---|---|---|---|

S_{5} | Symmetric group on 5 letters; = PGL_{2}(𝔽_{5}) = Aut(A_{5}) = 5-cell symmetries; almost simple | S5 | 120,34 | 15T10 |

S_{3}×F_{5} | Direct product of S_{3} and F_{5}; = Aut(D_{15}) = Hol(C_{15}) | S3xF5 | 120,36 | 15T11 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{5}^{2}⋊C_{6} | The semidirect product of C_{5}^{2} and C_{6} acting faithfully | C5^2:C6 | 150,6 | 15T12 |

C_{5}^{2}⋊S_{3} | The semidirect product of C_{5}^{2} and S_{3} acting faithfully | C5^2:S3 | 150,5 | 15T13 |

15T14 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{3}×A_{5} | Direct product of C_{3} and A_{5}; = GL_{2}(𝔽_{4}) | C3xA5 | 180,19 | 15T15 |

15T16 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{5}^{2}⋊Dic_{3} | The semidirect product of C_{5}^{2} and Dic_{3} acting faithfully | C5^2:Dic3 | 300,23 | 15T17 |

C_{5}^{2}⋊D_{6} | The semidirect product of C_{5}^{2} and D_{6} acting faithfully | C5^2:D6 | 300,25 | 15T18 |

C_{5}^{2}⋊C_{12} | The semidirect product of C_{5}^{2} and C_{12} acting faithfully | C5^2:C12 | 300,24 | 15T19 |

Label | ID | Tr ID | ||
---|---|---|---|---|

A_{6} | Alternating group on 6 letters; = PSL_{2}(𝔽_{9}) = L_{2}(9); 3^{rd} non-abelian simple | A6 | 360,118 | 15T20 |

ΓL_{2}(𝔽_{4}) | Semilinear group on 𝔽_{4}^{2}; = C_{3}⋊S_{5} | GammaL(2,4) | 360,120 | 15T21 |

15T22 | ||||

S_{3}×A_{5} | Direct product of S_{3} and A_{5} | S3xA5 | 360,121 | 15T23 |

C_{3}×S_{5} | Direct product of C_{3} and S_{5} | C3xS5 | 360,119 | 15T24 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{5}×C_{5}^{2}⋊C_{3} | Direct product of C_{5} and C_{5}^{2}⋊C_{3}; = AΣL_{1}(𝔽_{125}) | C5xC5^2:C3 | 375,6 | 15T25 |

Label | ID | Tr ID | ||
---|---|---|---|---|

C_{3}^{4}⋊C_{5} | The semidirect product of C_{3}^{4} and C_{5} acting faithfully | C3^4:C5 | 405,15 | 15T26 |

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