Transitive groups of degree up to 15

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 Sn for n=(G:H), the index of H in G, called the transitive degree. Conversely, all transitive subgroups of Sn arise in this way. The transitive group database in GAP and Magma contains all transitive subgroups of Sn up to conjugacy for n≤31, numbered nTi (or Tn,i).

The table below lists all transitive groups with n≤15. See also the full table with n≤31 and the smallest transitive degree table (n≤120).

On 1 point

Groups of order 1

LabelIDTr ID
C1Trivial groupC11,11T1

On 2 points

Groups of order 2

LabelIDTr ID
C2Cyclic groupC22,12T1

On 3 points

Groups of order 3

LabelIDTr ID
C3Cyclic group; = A3 = triangle rotationsC33,13T1

Groups of order 6

LabelIDTr ID
S3Symmetric group on 3 letters; = D3 = GL2(𝔽2) = triangle symmetries = 1st non-abelian groupS36,13T2

On 4 points

Groups of order 4

LabelIDTr ID
C4Cyclic group; = square rotationsC44,14T1
C22Klein 4-group V4 = elementary abelian group of type [2,2]; = rectangle symmetriesC2^24,24T2

Groups of order 8

LabelIDTr ID
D4Dihedral group; = He2 = AΣL1(𝔽4) = 2+ 1+2 = square symmetriesD48,34T3

Groups of order 12

LabelIDTr ID
A4Alternating group on 4 letters; = PSL2(𝔽3) = L2(3) = tetrahedron rotationsA412,34T4

Groups of order 24

LabelIDTr ID
S4Symmetric group on 4 letters; = PGL2(𝔽3) = Aut(Q8) = Hol(C22) = tetrahedron symmetries = cube/octahedron rotationsS424,124T5

On 5 points

Groups of order 5

LabelIDTr ID
C5Cyclic group; = pentagon rotationsC55,15T1

Groups of order 10

LabelIDTr ID
D5Dihedral group; = pentagon symmetriesD510,15T2

Groups of order 20

LabelIDTr ID
F5Frobenius group; = C5C4 = AGL1(𝔽5) = Aut(D5) = Hol(C5) = Sz(2)F520,35T3

Groups of order 60

LabelIDTr ID
A5Alternating group on 5 letters; = SL2(𝔽4) = L2(5) = L2(4) = icosahedron/dodecahedron rotations; 1st non-abelian simpleA560,55T4

Groups of order 120

LabelIDTr ID
S5Symmetric group on 5 letters; = PGL2(𝔽5) = Aut(A5) = 5-cell symmetries; almost simpleS5120,345T5

On 6 points

Groups of order 6

LabelIDTr ID
C6Cyclic group; = hexagon rotationsC66,26T1
S3Symmetric group on 3 letters; = D3 = GL2(𝔽2) = triangle symmetries = 1st non-abelian groupS36,16T2

Groups of order 12

LabelIDTr ID
D6Dihedral group; = C2×S3 = hexagon symmetriesD612,46T3
A4Alternating group on 4 letters; = PSL2(𝔽3) = L2(3) = tetrahedron rotationsA412,36T4

Groups of order 18

LabelIDTr ID
C3×S3Direct product of C3 and S3; = U2(𝔽2)C3xS318,36T5

Groups of order 24

LabelIDTr ID
C2×A4Direct product of C2 and A4; = AΣL1(𝔽8)C2xA424,136T6
S4Symmetric group on 4 letters; = PGL2(𝔽3) = Aut(Q8) = Hol(C22) = tetrahedron symmetries = cube/octahedron rotationsS424,126T7
6T8

Groups of order 36

LabelIDTr ID
S32Direct product of S3 and S3; = Spin+4(𝔽2) = Hol(S3)S3^236,106T9
C32⋊C4The semidirect product of C32 and C4 acting faithfullyC3^2:C436,96T10

Groups of order 48

LabelIDTr ID
C2×S4Direct product of C2 and S4; = O3(𝔽3) = cube/octahedron symmetriesC2xS448,486T11

Groups of order 60

LabelIDTr ID
A5Alternating group on 5 letters; = SL2(𝔽4) = L2(5) = L2(4) = icosahedron/dodecahedron rotations; 1st non-abelian simpleA560,56T12

Groups of order 72

LabelIDTr ID
S3≀C2Wreath product of S3 by C2; = SO+4(𝔽2)S3wrC272,406T13

Groups of order 120

LabelIDTr ID
S5Symmetric group on 5 letters; = PGL2(𝔽5) = Aut(A5) = 5-cell symmetries; almost simpleS5120,346T14

Groups of order 360

LabelIDTr ID
A6Alternating group on 6 letters; = PSL2(𝔽9) = L2(9); 3rd non-abelian simpleA6360,1186T15

On 7 points

Groups of order 7

LabelIDTr ID
C7Cyclic groupC77,17T1

Groups of order 14

LabelIDTr ID
D7Dihedral groupD714,17T2

Groups of order 21

LabelIDTr ID
C7⋊C3The semidirect product of C7 and C3 acting faithfullyC7:C321,17T3

Groups of order 42

LabelIDTr ID
F7Frobenius group; = C7C6 = AGL1(𝔽7) = Aut(D7) = Hol(C7)F742,17T4

Groups of order 168

LabelIDTr ID
GL3(𝔽2)General linear group on 𝔽23; = Aut(C23) = L3(2) = L2(7); 2nd non-abelian simpleGL(3,2)168,427T5

On 8 points

Groups of order 8

LabelIDTr ID
C8Cyclic groupC88,18T1
C2×C4Abelian group of type [2,4]C2xC48,28T2
C23Elementary abelian group of type [2,2,2]C2^38,58T3
D4Dihedral group; = He2 = AΣL1(𝔽4) = 2+ 1+2 = square symmetriesD48,38T4
Q8Quaternion group; = C4.C2 = Dic2 = 2- 1+2Q88,48T5

Groups of order 16

LabelIDTr ID
D8Dihedral groupD816,78T6
M4(2)Modular maximal-cyclic group; = C83C2M4(2)16,68T7
SD16Semidihedral group; = Q8C2 = QD16SD1616,88T8
C2×D4Direct product of C2 and D4C2xD416,118T9
C22⋊C4The semidirect product of C22 and C4 acting via C4/C2=C2C2^2:C416,38T10
C4○D4Pauli group = central product of C4 and D4C4oD416,138T11

Groups of order 24

LabelIDTr ID
SL2(𝔽3)Special linear group on 𝔽32; = Q8C3 = 2T = <2,3,3> = 1st non-monomial groupSL(2,3)24,38T12
C2×A4Direct product of C2 and A4; = AΣL1(𝔽8)C2xA424,138T13
S4Symmetric group on 4 letters; = PGL2(𝔽3) = Aut(Q8) = Hol(C22) = tetrahedron symmetries = cube/octahedron rotationsS424,128T14

Groups of order 32

LabelIDTr ID
C8⋊C22The semidirect product of C8 and C22 acting faithfully; = Aut(D8) = Hol(C8)C8:C2^232,438T15
C4.D41st non-split extension by C4 of D4 acting via D4/C22=C2C4.D432,78T16
C4≀C2Wreath product of C4 by C2C4wrC232,118T17
C22≀C2Wreath product of C22 by C2C2^2wrC232,278T18
C23⋊C4The semidirect product of C23 and C4 acting faithfullyC2^3:C432,68T19
8T20
8T21
2+ 1+4Extraspecial group; = D4D4ES+(2,2)32,498T22

Groups of order 48

LabelIDTr ID
GL2(𝔽3)General linear group on 𝔽32; = Q8S3 = Aut(C32) = 2O = <2,3,4>GL(2,3)48,298T23
C2×S4Direct product of C2 and S4; = O3(𝔽3) = cube/octahedron symmetriesC2xS448,488T24

Groups of order 56

LabelIDTr ID
F8Frobenius group; = C23C7 = AGL1(𝔽8)F856,118T25

Groups of order 64

LabelIDTr ID
D44D43rd semidirect product of D4 and D4 acting via D4/C22=C2; = Hol(D4)D4:4D464,1348T26
C2≀C4Wreath product of C2 by C4; = AΣL1(𝔽16)C2wrC464,328T27
8T28
C2≀C22Wreath product of C2 by C22; = Hol(C2×C4)C2wrC2^264,1388T29
C42⋊C42nd semidirect product of C42 and C4 acting faithfullyC4^2:C464,348T30
C2≀C22Wreath product of C2 by C22; = Hol(C2×C4)C2wrC2^264,1388T31

Groups of order 96

LabelIDTr ID
C23⋊A42nd semidirect product of C23 and A4 acting faithfullyC2^3:A496,2048T32
C24⋊C61st semidirect product of C24 and C6 acting faithfullyC2^4:C696,708T33
C22⋊S4The semidirect product of C22 and S4 acting via S4/C22=S3C2^2:S496,2278T34

Groups of order 128

LabelIDTr ID
C2≀D4Wreath product of C2 by D4C2wrD4128,9288T35

Groups of order 168

LabelIDTr ID
AΓL1(𝔽8)Affine semilinear group on 𝔽81; = F8C3 = Aut(F8)AGammaL(1,8)168,438T36
GL3(𝔽2)General linear group on 𝔽23; = Aut(C23) = L3(2) = L2(7); 2nd non-abelian simpleGL(3,2)168,428T37

Groups of order 192

LabelIDTr ID
C2≀A4Wreath product of C2 by A4C2wrA4192,2018T38
C23⋊S42nd semidirect product of C23 and S4 acting faithfully; = Aut(C22×C4)C2^3:S4192,14938T39
Q82S42nd semidirect product of Q8 and S4 acting via S4/C22=S3; = Hol(Q8)Q8:2S4192,14948T40
C24⋊D61st semidirect product of C24 and D6 acting faithfully; = Aut(C2×Q8)C2^4:D6192,9558T41

Groups of order 288

LabelIDTr ID
A4≀C2Wreath product of A4 by C2A4wrC2288,10258T42

Groups of order 336

LabelIDTr ID
PGL2(𝔽7)Projective linear group on 𝔽72; = GL3(𝔽2)C2 = Aut(GL3(𝔽2)); almost simplePGL(2,7)336,2088T43

On 9 points

Groups of order 9

LabelIDTr ID
C9Cyclic groupC99,19T1
C32Elementary abelian group of type [3,3]C3^29,29T2

Groups of order 18

LabelIDTr ID
D9Dihedral groupD918,19T3
C3×S3Direct product of C3 and S3; = U2(𝔽2)C3xS318,39T4
C3⋊S3The semidirect product of C3 and S3 acting via S3/C3=C2C3:S318,49T5

Groups of order 27

LabelIDTr ID
3- 1+2Extraspecial groupES-(3,1)27,49T6
He3Heisenberg group; = C32C3 = 3+ 1+2He327,39T7

Groups of order 36

LabelIDTr ID
S32Direct product of S3 and S3; = Spin+4(𝔽2) = Hol(S3)S3^236,109T8
C32⋊C4The semidirect product of C32 and C4 acting faithfullyC3^2:C436,99T9

Groups of order 54

LabelIDTr ID
C9⋊C6The semidirect product of C9 and C6 acting faithfully; = Aut(D9) = Hol(C9)C9:C654,69T10
C32⋊C6The semidirect product of C32 and C6 acting faithfullyC3^2:C654,59T11
He3⋊C22nd semidirect product of He3 and C2 acting faithfully; = Aut(3- 1+2)He3:C254,89T12
C32⋊C6The semidirect product of C32 and C6 acting faithfullyC3^2:C654,59T13

Groups of order 72

LabelIDTr ID
PSU3(𝔽2)Projective special unitary group on 𝔽23; = C32Q8 = M9PSU(3,2)72,419T14
F9Frobenius group; = C32C8 = AGL1(𝔽9)F972,399T15
S3≀C2Wreath product of S3 by C2; = SO+4(𝔽2)S3wrC272,409T16

Groups of order 81

LabelIDTr ID
C3≀C3Wreath product of C3 by C3; = AΣL1(𝔽27)C3wrC381,79T17

Groups of order 108

LabelIDTr ID
C32⋊D6The semidirect product of C32 and D6 acting faithfullyC3^2:D6108,179T18

Groups of order 144

LabelIDTr ID
AΓL1(𝔽9)Affine semilinear group on 𝔽91; = F9C2 = Aut(C32⋊C4)AGammaL(1,9)144,1829T19

Groups of order 162

LabelIDTr ID
C3≀S3Wreath product of C3 by S3C3wrS3162,109T20
C33⋊S32nd semidirect product of C33 and S3 acting faithfullyC3^3:S3162,199T21
C33⋊C61st semidirect product of C33 and C6 acting faithfullyC3^3:C6162,119T22

Groups of order 216

LabelIDTr ID
ASL2(𝔽3)Hessian group = Affine special linear group on 𝔽32; = PSU3(𝔽2)C3ASL(2,3)216,1539T23

Groups of order 324

LabelIDTr ID
He3⋊D6The semidirect product of He3 and D6 acting faithfullyHe3:D6324,399T24
C33⋊A4The semidirect product of C33 and A4 acting faithfullyC3^3:A4324,1609T25

Groups of order 432

LabelIDTr ID
AGL2(𝔽3)Affine linear group on 𝔽32; = PSU3(𝔽2)S3 = Aut(C3⋊S3) = Hol(C32)AGL(2,3)432,7349T26

On 10 points

Groups of order 10

LabelIDTr ID
C10Cyclic groupC1010,210T1
D5Dihedral group; = pentagon symmetriesD510,110T2

Groups of order 20

LabelIDTr ID
D10Dihedral group; = C2×D5D1020,410T3
F5Frobenius group; = C5C4 = AGL1(𝔽5) = Aut(D5) = Hol(C5) = Sz(2)F520,310T4

Groups of order 40

LabelIDTr ID
C2×F5Direct product of C2 and F5; = Aut(D10) = Hol(C10)C2xF540,1210T5

Groups of order 50

LabelIDTr ID
C5×D5Direct product of C5 and D5; = AΣL1(𝔽25)C5xD550,310T6

Groups of order 60

LabelIDTr ID
A5Alternating group on 5 letters; = SL2(𝔽4) = L2(5) = L2(4) = icosahedron/dodecahedron rotations; 1st non-abelian simpleA560,510T7

Groups of order 80

LabelIDTr ID
C24⋊C5The semidirect product of C24 and C5 acting faithfullyC2^4:C580,4910T8

Groups of order 100

LabelIDTr ID
D52Direct product of D5 and D5D5^2100,1310T9
C52⋊C44th semidirect product of C52 and C4 acting faithfullyC5^2:C4100,1210T10

Groups of order 120

LabelIDTr ID
C2×A5Direct product of C2 and A5; = icosahedron/dodecahedron symmetriesC2xA5120,3510T11
S5Symmetric group on 5 letters; = PGL2(𝔽5) = Aut(A5) = 5-cell symmetries; almost simpleS5120,3410T12
10T13

Groups of order 160

LabelIDTr ID
C2×C24⋊C5Direct product of C2 and C24⋊C5; = AΣL1(𝔽32)C2xC2^4:C5160,23510T14
C24⋊D5The semidirect product of C24 and D5 acting faithfullyC2^4:D5160,23410T15
10T16

Groups of order 200

LabelIDTr ID
D5⋊F5The semidirect product of D5 and F5 acting via F5/D5=C2; = Hol(D5)D5:F5200,4210T17
C52⋊C8The semidirect product of C52 and C8 acting faithfullyC5^2:C8200,4010T18
D5≀C2Wreath product of D5 by C2D5wrC2200,4310T19
C52⋊Q8The semidirect product of C52 and Q8 acting faithfullyC5^2:Q8200,4410T20
D5≀C2Wreath product of D5 by C2D5wrC2200,4310T21

Groups of order 240

LabelIDTr ID
C2×S5Direct product of C2 and S5; = O3(𝔽5)C2xS5240,18910T22

Groups of order 320

LabelIDTr ID
C2×C24⋊D5Direct product of C2 and C24⋊D5C2xC2^4:D5320,163610T23
C24⋊F5The semidirect product of C24 and F5 acting faithfullyC2^4:F5320,163510T24
10T25

Groups of order 360

LabelIDTr ID
A6Alternating group on 6 letters; = PSL2(𝔽9) = L2(9); 3rd non-abelian simpleA6360,11810T26

Groups of order 400

LabelIDTr ID
D5≀C2⋊C2The semidirect product of D5≀C2 and C2 acting faithfullyD5wrC2:C2400,20710T27
C52⋊M4(2)The semidirect product of C52 and M4(2) acting faithfullyC5^2:M4(2)400,20610T28

On 11 points

Groups of order 11

LabelIDTr ID
C11Cyclic groupC1111,111T1

Groups of order 22

LabelIDTr ID
D11Dihedral groupD1122,111T2

Groups of order 55

LabelIDTr ID
C11⋊C5The semidirect product of C11 and C5 acting faithfullyC11:C555,111T3

Groups of order 110

LabelIDTr ID
F11Frobenius group; = C11C10 = AGL1(𝔽11) = Aut(D11) = Hol(C11)F11110,111T4

On 12 points

Groups of order 12

LabelIDTr ID
C12Cyclic groupC1212,212T1
C2×C6Abelian group of type [2,6]C2xC612,512T2
D6Dihedral group; = C2×S3 = hexagon symmetriesD612,412T3
A4Alternating group on 4 letters; = PSL2(𝔽3) = L2(3) = tetrahedron rotationsA412,312T4
Dic3Dicyclic group; = C3C4Dic312,112T5

Groups of order 24

LabelIDTr ID
C2×A4Direct product of C2 and A4; = AΣL1(𝔽8)C2xA424,1312T6
12T7
S4Symmetric group on 4 letters; = PGL2(𝔽3) = Aut(Q8) = Hol(C22) = tetrahedron symmetries = cube/octahedron rotationsS424,1212T8
12T9
C22×S3Direct product of C22 and S3C2^2xS324,1412T10
C4×S3Direct product of C4 and S3C4xS324,512T11
D12Dihedral groupD1224,612T12
C3⋊D4The semidirect product of C3 and D4 acting via D4/C22=C2C3:D424,812T13
C3×D4Direct product of C3 and D4C3xD424,1012T14
C3⋊D4The semidirect product of C3 and D4 acting via D4/C22=C2C3:D424,812T15

Groups of order 36

LabelIDTr ID
S32Direct product of S3 and S3; = Spin+4(𝔽2) = Hol(S3)S3^236,1012T16
C32⋊C4The semidirect product of C32 and C4 acting faithfullyC3^2:C436,912T17
S3×C6Direct product of C6 and S3S3xC636,1212T18
C3×Dic3Direct product of C3 and Dic3C3xDic336,612T19
C3×A4Direct product of C3 and A4C3xA436,1112T20

Groups of order 48

LabelIDTr ID
C2×S4Direct product of C2 and S4; = O3(𝔽3) = cube/octahedron symmetriesC2xS448,4812T21
12T22
12T23
12T24
C22×A4Direct product of C22 and A4C2^2xA448,4912T25
12T26
A4⋊C4The semidirect product of A4 and C4 acting via C4/C2=C2; = SL2(ℤ/4ℤ)A4:C448,3012T27
S3×D4Direct product of S3 and D4; = Aut(D12) = Hol(C12)S3xD448,3812T28
C4×A4Direct product of C4 and A4C4xA448,3112T29
A4⋊C4The semidirect product of A4 and C4 acting via C4/C2=C2; = SL2(ℤ/4ℤ)A4:C448,3012T30
C42⋊C3The semidirect product of C42 and C3 acting faithfullyC4^2:C348,312T31
C22⋊A4The semidirect product of C22 and A4 acting via A4/C22=C3C2^2:A448,5012T32

Groups of order 60

LabelIDTr ID
A5Alternating group on 5 letters; = SL2(𝔽4) = L2(5) = L2(4) = icosahedron/dodecahedron rotations; 1st non-abelian simpleA560,512T33

Groups of order 72

LabelIDTr ID
S3≀C2Wreath product of S3 by C2; = SO+4(𝔽2)S3wrC272,4012T34
12T35
12T36
C2×S32Direct product of C2, S3 and S3C2xS3^272,4612T37
C3⋊D12The semidirect product of C3 and D12 acting via D12/D6=C2C3:D1272,2312T38
C6.D62nd non-split extension by C6 of D6 acting via D6/S3=C2C6.D672,2112T39
C2×C32⋊C4Direct product of C2 and C32⋊C4C2xC3^2:C472,4512T40
12T41
C3×C3⋊D4Direct product of C3 and C3⋊D4C3xC3:D472,3012T42
S3×A4Direct product of S3 and A4S3xA472,4412T43
C3⋊S4The semidirect product of C3 and S4 acting via S4/A4=C2C3:S472,4312T44
C3×S4Direct product of C3 and S4C3xS472,4212T45
F9Frobenius group; = C32C8 = AGL1(𝔽9)F972,3912T46
PSU3(𝔽2)Projective special unitary group on 𝔽23; = C32Q8 = M9PSU(3,2)72,4112T47

Groups of order 96

LabelIDTr ID
C22×S4Direct product of C22 and S4C2^2xS496,22612T48
A4⋊D4The semidirect product of A4 and D4 acting via D4/C22=C2; = Aut(C42) = GL2(ℤ/4ℤ)A4:D496,19512T49
12T50
D4×A4Direct product of D4 and A4D4xA496,19712T51
A4⋊D4The semidirect product of A4 and D4 acting via D4/C22=C2; = Aut(C42) = GL2(ℤ/4ℤ)A4:D496,19512T52
C4×S4Direct product of C4 and S4C4xS496,18612T53
C4⋊S4The semidirect product of C4 and S4 acting via S4/A4=C2C4:S496,18712T54
C2×C42⋊C3Direct product of C2 and C42⋊C3C2xC4^2:C396,6812T55
C2×C22⋊A4Direct product of C2 and C22⋊A4C2xC2^2:A496,22912T56
C23.3A41st non-split extension by C23 of A4 acting via A4/C22=C3C2^3.3A496,312T57
C24⋊C61st semidirect product of C24 and C6 acting faithfullyC2^4:C696,7012T58
12T59
C23.A42nd non-split extension by C23 of A4 acting faithfullyC2^3.A496,7212T60
12T61
C42⋊S3The semidirect product of C42 and S3 acting faithfullyC4^2:S396,6412T62
12T63
12T64
12T65
C22⋊S4The semidirect product of C22 and S4 acting via S4/C22=S3C2^2:S496,22712T66
12T67
12T68
12T69

Groups of order 108

LabelIDTr ID
C3×S32Direct product of C3, S3 and S3C3xS3^2108,3812T70
C324D6The semidirect product of C32 and D6 acting via D6/C3=C22C3^2:4D6108,4012T71
C33⋊C42nd semidirect product of C33 and C4 acting faithfullyC3^3:C4108,3712T72
C3×C32⋊C4Direct product of C3 and C32⋊C4C3xC3^2:C4108,3612T73

Groups of order 120

LabelIDTr ID
S5Symmetric group on 5 letters; = PGL2(𝔽5) = Aut(A5) = 5-cell symmetries; almost simpleS5120,3412T74
C2×A5Direct product of C2 and A5; = icosahedron/dodecahedron symmetriesC2xA5120,3512T75
12T76

Groups of order 144

LabelIDTr ID
C2×S3≀C2Direct product of C2 and S3≀C2C2xS3wrC2144,18612T77
12T78
(S32)⋊C4The semidirect product of S32 and C4 acting via C4/C2=C2(S3^2):C4144,11512T79
12T80
Dic3⋊D62nd semidirect product of Dic3 and D6 acting via D6/S3=C2; = Hol(Dic3)Dic3:D6144,15412T81
C62⋊C41st semidirect product of C62 and C4 acting faithfullyC6^2:C4144,13612T82
S3×S4Direct product of S3 and S4; = Hol(C2×C6)S3xS4144,18312T83
AΓL1(𝔽9)Affine semilinear group on 𝔽91; = F9C2 = Aut(C32⋊C4)AGammaL(1,9)144,18212T84
A42Direct product of A4 and A4; = PΩ+4(𝔽3)A4^2144,18412T85

Groups of order 192

LabelIDTr ID
D4×S4Direct product of D4 and S4D4xS4192,147212T86
C2×C24⋊C6Direct product of C2 and C24⋊C6C2xC2^4:C6192,100012T87
12T88
C2×C23.A4Direct product of C2 and C23.A4C2xC2^3.A4192,100212T89
C22×C22⋊A4Direct product of C22 and C22⋊A4C2^2xC2^2:A4192,154012T90
C24.2A42nd non-split extension by C24 of A4 acting faithfullyC2^4.2A4192,19712T91
C2×C23.A4Direct product of C2 and C23.A4C2xC2^3.A4192,100212T92
C24.2A42nd non-split extension by C24 of A4 acting faithfullyC2^4.2A4192,19712T93
C4×C42⋊C3Direct product of C4 and C42⋊C3C4xC4^2:C3192,18812T94
C2×C42⋊S3Direct product of C2 and C42⋊S3C2xC4^2:S3192,94412T95
12T96
12T97
C23.9S43rd non-split extension by C23 of S4 acting via S4/C22=S3C2^3.9S4192,18212T98
C24⋊C121st semidirect product of C24 and C12 acting via C12/C2=C6C2^4:C12192,19112T99
C2×C22⋊S4Direct product of C2 and C22⋊S4C2xC2^2:S4192,153812T100
12T101
C244Dic33rd semidirect product of C24 and Dic3 acting via Dic3/C2=S3C2^4:4Dic3192,149512T102
C2×C22⋊S4Direct product of C2 and C22⋊S4C2xC2^2:S4192,153812T103
C232D4⋊C3The semidirect product of C232D4 and C3 acting faithfullyC2^3:2D4:C3192,19412T104
C24⋊C121st semidirect product of C24 and C12 acting via C12/C2=C6C2^4:C12192,19112T105
C2×C22⋊S4Direct product of C2 and C22⋊S4C2xC2^2:S4192,153812T106
C244Dic33rd semidirect product of C24 and Dic3 acting via Dic3/C2=S3C2^4:4Dic3192,149512T107
C24⋊D61st semidirect product of C24 and D6 acting faithfully; = Aut(C2×Q8)C2^4:D6192,95512T108
12T109
12T110
12T111
C42⋊D6The semidirect product of C42 and D6 acting faithfullyC4^2:D6192,95612T112
12T113
12T114
12T115

Groups of order 216

LabelIDTr ID
C33⋊D42nd semidirect product of C33 and D4 acting faithfullyC3^3:D4216,15812T116
S33Direct product of S3, S3 and S3; = Hol(C3×S3)S3^3216,16212T117
C322D12The semidirect product of C32 and D12 acting via D12/C3=D4C3^2:2D12216,15912T118
S3×C32⋊C4Direct product of S3 and C32⋊C4S3xC3^2:C4216,15612T119
C33⋊D42nd semidirect product of C33 and D4 acting faithfullyC3^3:D4216,15812T120
C3×S3≀C2Direct product of C3 and S3≀C2C3xS3wrC2216,15712T121
ASL2(𝔽3)Hessian group = Affine special linear group on 𝔽32; = PSU3(𝔽2)C3ASL(2,3)216,15312T122

Groups of order 240

LabelIDTr ID
C2×S5Direct product of C2 and S5; = O3(𝔽5)C2xS5240,18912T123
A5⋊C4The semidirect product of A5 and C4 acting via C4/C2=C2A5:C4240,9112T124

Groups of order 288

LabelIDTr ID
D6≀C2Wreath product of D6 by C2D6wrC2288,88912T125
A4≀C2Wreath product of A4 by C2A4wrC2288,102512T126
PSO4+ (𝔽3)Projective special orthogonal group of + type on 𝔽34; = A4S4 = Hol(A4)PSO+(4,3)288,102612T127
A4≀C2Wreath product of A4 by C2A4wrC2288,102512T128
12T129

Groups of order 324

LabelIDTr ID
C3×C324D6Direct product of C3 and C324D6C3xC3^2:4D6324,16712T130
C3×C33⋊C4Direct product of C3 and C33⋊C4; = AΣL1(𝔽81)C3xC3^3:C4324,16212T131
C33⋊A4The semidirect product of C33 and A4 acting faithfullyC3^3:A4324,16012T132
12T133

Groups of order 432

LabelIDTr ID
S3×S3≀C2Direct product of S3 and S3≀C2S3xS3wrC2432,74112T156
AGL2(𝔽3)Affine linear group on 𝔽32; = PSU3(𝔽2)S3 = Aut(C3⋊S3) = Hol(C32)AGL(2,3)432,73412T157

On 13 points

Groups of order 13

LabelIDTr ID
C13Cyclic groupC1313,113T1

Groups of order 26

LabelIDTr ID
D13Dihedral groupD1326,113T2

Groups of order 39

LabelIDTr ID
C13⋊C3The semidirect product of C13 and C3 acting faithfullyC13:C339,113T3

Groups of order 52

LabelIDTr ID
C13⋊C4The semidirect product of C13 and C4 acting faithfullyC13:C452,313T4

Groups of order 78

LabelIDTr ID
C13⋊C6The semidirect product of C13 and C6 acting faithfullyC13:C678,113T5

Groups of order 156

LabelIDTr ID
F13Frobenius group; = C13C12 = AGL1(𝔽13) = Aut(D13) = Hol(C13)F13156,713T6

On 14 points

Groups of order 14

LabelIDTr ID
C14Cyclic groupC1414,214T1
D7Dihedral groupD714,114T2

Groups of order 28

LabelIDTr ID
D14Dihedral group; = C2×D7D1428,314T3

Groups of order 42

LabelIDTr ID
F7Frobenius group; = C7C6 = AGL1(𝔽7) = Aut(D7) = Hol(C7)F742,114T4
C2×C7⋊C3Direct product of C2 and C7⋊C3C2xC7:C342,214T5

Groups of order 56

LabelIDTr ID
F8Frobenius group; = C23C7 = AGL1(𝔽8)F856,1114T6

Groups of order 84

LabelIDTr ID
C2×F7Direct product of C2 and F7; = Aut(D14) = Hol(C14)C2xF784,714T7

Groups of order 98

LabelIDTr ID
C7×D7Direct product of C7 and D7; = AΣL1(𝔽49)C7xD798,314T8

Groups of order 112

LabelIDTr ID
C2×F8Direct product of C2 and F8C2xF8112,4114T9

Groups of order 168

LabelIDTr ID
GL3(𝔽2)General linear group on 𝔽23; = Aut(C23) = L3(2) = L2(7); 2nd non-abelian simpleGL(3,2)168,4214T10
AΓL1(𝔽8)Affine semilinear group on 𝔽81; = F8C3 = Aut(F8)AGammaL(1,8)168,4314T11

Groups of order 196

LabelIDTr ID
C72⋊C4The semidirect product of C72 and C4 acting faithfullyC7^2:C4196,814T12
D72Direct product of D7 and D7D7^2196,914T13

Groups of order 294

LabelIDTr ID
C74F72nd semidirect product of C7 and F7 acting via F7/D7=C3C7:4F7294,1214T14
C72⋊S3The semidirect product of C72 and S3 acting faithfullyC7^2:S3294,714T15

Groups of order 336

LabelIDTr ID
PGL2(𝔽7)Projective linear group on 𝔽72; = GL3(𝔽2)C2 = Aut(GL3(𝔽2)); almost simplePGL(2,7)336,20814T16
C2×GL3(𝔽2)Direct product of C2 and GL3(𝔽2)C2xGL(3,2)336,20914T17
C2×AΓL1(𝔽8)Direct product of C2 and AΓL1(𝔽8)C2xAGammaL(1,8)336,21014T18
C2×GL3(𝔽2)Direct product of C2 and GL3(𝔽2)C2xGL(3,2)336,20914T19

Groups of order 392

LabelIDTr ID
D7≀C2Wreath product of D7 by C2D7wrC2392,3714T20

Groups of order 448

LabelIDTr ID
C23⋊F82nd semidirect product of C23 and F8 acting via F8/C23=C7C2^3:F8448,139414T21

On 15 points

Groups of order 15

LabelIDTr ID
C15Cyclic groupC1515,115T1

Groups of order 30

LabelIDTr ID
D15Dihedral groupD1530,315T2
C3×D5Direct product of C3 and D5C3xD530,215T3
C5×S3Direct product of C5 and S3C5xS330,115T4

Groups of order 60

LabelIDTr ID
A5Alternating group on 5 letters; = SL2(𝔽4) = L2(5) = L2(4) = icosahedron/dodecahedron rotations; 1st non-abelian simpleA560,515T5
C3⋊F5The semidirect product of C3 and F5 acting via F5/D5=C2C3:F560,715T6
S3×D5Direct product of S3 and D5S3xD560,815T7
C3×F5Direct product of C3 and F5C3xF560,615T8

Groups of order 75

LabelIDTr ID
C52⋊C3The semidirect product of C52 and C3 acting faithfullyC5^2:C375,215T9

Groups of order 120

LabelIDTr ID
S5Symmetric group on 5 letters; = PGL2(𝔽5) = Aut(A5) = 5-cell symmetries; almost simpleS5120,3415T10
S3×F5Direct product of S3 and F5; = Aut(D15) = Hol(C15)S3xF5120,3615T11

Groups of order 150

LabelIDTr ID
C52⋊C6The semidirect product of C52 and C6 acting faithfullyC5^2:C6150,615T12
C52⋊S3The semidirect product of C52 and S3 acting faithfullyC5^2:S3150,515T13
15T14

Groups of order 180

LabelIDTr ID
C3×A5Direct product of C3 and A5; = GL2(𝔽4)C3xA5180,1915T15
15T16

Groups of order 300

LabelIDTr ID
C52⋊Dic3The semidirect product of C52 and Dic3 acting faithfullyC5^2:Dic3300,2315T17
C52⋊D6The semidirect product of C52 and D6 acting faithfullyC5^2:D6300,2515T18
C52⋊C12The semidirect product of C52 and C12 acting faithfullyC5^2:C12300,2415T19

Groups of order 360

LabelIDTr ID
A6Alternating group on 6 letters; = PSL2(𝔽9) = L2(9); 3rd non-abelian simpleA6360,11815T20
ΓL2(𝔽4)Semilinear group on 𝔽42; = C3S5GammaL(2,4)360,12015T21
15T22
S3×A5Direct product of S3 and A5S3xA5360,12115T23
C3×S5Direct product of C3 and S5C3xS5360,11915T24

Groups of order 375

LabelIDTr ID
C5×C52⋊C3Direct product of C5 and C52⋊C3; = AΣL1(𝔽125)C5xC5^2:C3375,615T25

Groups of order 405

LabelIDTr ID
C34⋊C5The semidirect product of C34 and C5 acting faithfullyC3^4:C5405,1515T26
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