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 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 nTi (or T_n,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

		Label	ID	Tr ID
C₁	Trivial group	C1	1,1	1T1

On 2 points

Groups of order 2

		Label	ID	Tr ID
C₂	Cyclic group	C2	2,1	2T1

On 3 points

Groups of order 3

		Label	ID	Tr ID
C₃	Cyclic group; = A₃ = triangle rotations	C3	3,1	3T1

Groups of order 6

		Label	ID	Tr ID
S₃	Symmetric group on 3 letters; = D₃ = GL₂(𝔽₂) = triangle symmetries = 1^st non-abelian group	S3	6,1	3T2

On 4 points

Groups of order 4

		Label	ID	Tr ID
C₄	Cyclic group; = square rotations	C4	4,1	4T1
C₂²	Klein 4-group V₄ = elementary abelian group of type [2,2]; = rectangle symmetries	C2^2	4,2	4T2

Groups of order 8

		Label	ID	Tr ID
D₄	Dihedral group; = He₂ = AΣL₁(𝔽₄) = 2₊¹⁺² = square symmetries	D4	8,3	4T3

Groups of order 12

		Label	ID	Tr ID
A₄	Alternating group on 4 letters; = PSL₂(𝔽₃) = L₂(3) = tetrahedron rotations	A4	12,3	4T4

Groups of order 24

		Label	ID	Tr ID
S₄	Symmetric group on 4 letters; = PGL₂(𝔽₃) = Aut(Q₈) = Hol(C₂²) = tetrahedron symmetries = cube/octahedron rotations	S4	24,12	4T5

On 5 points

Groups of order 5

		Label	ID	Tr ID
C₅	Cyclic group; = pentagon rotations	C5	5,1	5T1

Groups of order 10

		Label	ID	Tr ID
D₅	Dihedral group; = pentagon symmetries	D5	10,1	5T2

Groups of order 20

		Label	ID	Tr ID
F₅	Frobenius group; = C₅ ⋊C₄ = AGL₁(𝔽₅) = Aut(D₅) = Hol(C₅) = Sz(2)	F5	20,3	5T3

Groups of order 60

		Label	ID	Tr ID
A₅	Alternating group on 5 letters; = SL₂(𝔽₄) = L₂(5) = L₂(4) = icosahedron/dodecahedron rotations; 1^st non-abelian simple	A5	60,5	5T4

Groups of order 120

		Label	ID	Tr ID
S₅	Symmetric group on 5 letters; = PGL₂(𝔽₅) = Aut(A₅) = 5-cell symmetries; almost simple	S5	120,34	5T5

On 6 points

Groups of order 6

		Label	ID	Tr ID
C₆	Cyclic group; = hexagon rotations	C6	6,2	6T1
S₃	Symmetric group on 3 letters; = D₃ = GL₂(𝔽₂) = triangle symmetries = 1^st non-abelian group	S3	6,1	6T2

Groups of order 12

		Label	ID	Tr ID
D₆	Dihedral group; = C₂×S₃ = hexagon symmetries	D6	12,4	6T3
A₄	Alternating group on 4 letters; = PSL₂(𝔽₃) = L₂(3) = tetrahedron rotations	A4	12,3	6T4

Groups of order 18

		Label	ID	Tr ID
C₃×S₃	Direct product of C₃ and S₃; = U₂(𝔽₂)	C3xS3	18,3	6T5

Groups of order 24

		Label	ID	Tr ID
C₂×A₄	Direct product of C₂ and A₄; = AΣL₁(𝔽₈)	C2xA4	24,13	6T6
S₄	Symmetric group on 4 letters; = PGL₂(𝔽₃) = Aut(Q₈) = Hol(C₂²) = tetrahedron symmetries = cube/octahedron rotations	S4	24,12	6T7
				6T8

Groups of order 36

		Label	ID	Tr ID
S₃²	Direct product of S₃ and S₃; = Spin+₄(𝔽₂) = Hol(S₃)	S3^2	36,10	6T9
C₃²⋊C₄	The semidirect product of C₃² and C₄ acting faithfully	C3^2:C4	36,9	6T10

Groups of order 48

		Label	ID	Tr ID
C₂×S₄	Direct product of C₂ and S₄; = O₃(𝔽₃) = cube/octahedron symmetries	C2xS4	48,48	6T11

Groups of order 60

		Label	ID	Tr ID
A₅	Alternating group on 5 letters; = SL₂(𝔽₄) = L₂(5) = L₂(4) = icosahedron/dodecahedron rotations; 1^st non-abelian simple	A5	60,5	6T12

Groups of order 72

		Label	ID	Tr ID
S₃≀C₂	Wreath product of S₃ by C₂; = SO+₄(𝔽₂)	S3wrC2	72,40	6T13

Groups of order 120

		Label	ID	Tr ID
S₅	Symmetric group on 5 letters; = PGL₂(𝔽₅) = Aut(A₅) = 5-cell symmetries; almost simple	S5	120,34	6T14

Groups of order 360

		Label	ID	Tr ID
A₆	Alternating group on 6 letters; = PSL₂(𝔽₉) = L₂(9); 3^rd non-abelian simple	A6	360,118	6T15

On 7 points

Groups of order 7

		Label	ID	Tr ID
C₇	Cyclic group	C7	7,1	7T1

Groups of order 14

		Label	ID	Tr ID
D₇	Dihedral group	D7	14,1	7T2

Groups of order 21

		Label	ID	Tr ID
C₇⋊C₃	The semidirect product of C₇ and C₃ acting faithfully	C7:C3	21,1	7T3

Groups of order 42

		Label	ID	Tr ID
F₇	Frobenius group; = C₇ ⋊C₆ = AGL₁(𝔽₇) = Aut(D₇) = Hol(C₇)	F7	42,1	7T4

Groups of order 168

		Label	ID	Tr ID
GL₃(𝔽₂)	General linear group on 𝔽₂³; = Aut(C₂³) = L₃(2) = L₂(7); 2^nd non-abelian simple	GL(3,2)	168,42	7T5

On 8 points

Groups of order 8

		Label	ID	Tr ID
C₈	Cyclic group	C8	8,1	8T1
C₂×C₄	Abelian group of type [2,4]	C2xC4	8,2	8T2
C₂³	Elementary abelian group of type [2,2,2]	C2^3	8,5	8T3
D₄	Dihedral group; = He₂ = AΣL₁(𝔽₄) = 2₊¹⁺² = square symmetries	D4	8,3	8T4
Q₈	Quaternion group; = C₄ .C₂ = Dic₂ = 2_-¹⁺²	Q8	8,4	8T5

Groups of order 16

		Label	ID	Tr ID
D₈	Dihedral group	D8	16,7	8T6
M₄(2)	Modular maximal-cyclic group; = C₈ ⋊₃ C₂	M4(2)	16,6	8T7
SD₁₆	Semidihedral group; = Q₈ ⋊C₂ = QD₁₆	SD16	16,8	8T8
C₂×D₄	Direct product of C₂ and D₄	C2xD4	16,11	8T9
C₂²⋊C₄	The semidirect product of C₂² and C₄ acting via C₄/C₂=C₂	C2^2:C4	16,3	8T10
C₄○D₄	Pauli group = central product of C₄ and D₄	C4oD4	16,13	8T11

Groups of order 24

		Label	ID	Tr ID
SL₂(𝔽₃)	Special linear group on 𝔽₃²; = Q₈ ⋊C₃ = 2T = <2,3,3> = 1^st non-monomial group	SL(2,3)	24,3	8T12
C₂×A₄	Direct product of C₂ and A₄; = AΣL₁(𝔽₈)	C2xA4	24,13	8T13
S₄	Symmetric group on 4 letters; = PGL₂(𝔽₃) = Aut(Q₈) = Hol(C₂²) = tetrahedron symmetries = cube/octahedron rotations	S4	24,12	8T14

Groups of order 32

		Label	ID	Tr ID
C₈⋊C₂²	The semidirect product of C₈ and C₂² acting faithfully; = Aut(D₈) = Hol(C₈)	C8:C2^2	32,43	8T15
C₄.D₄	1^st non-split extension by C₄ of D₄ acting via D₄/C₂²=C₂	C4.D4	32,7	8T16
C₄≀C₂	Wreath product of C₄ by C₂	C4wrC2	32,11	8T17
C₂²≀C₂	Wreath product of C₂² by C₂	C2^2wrC2	32,27	8T18
C₂³⋊C₄	The semidirect product of C₂³ and C₄ acting faithfully	C2^3:C4	32,6	8T19
				8T20
				8T21
2₊¹⁺⁴	Extraspecial group; = D₄ ○D₄	ES+(2,2)	32,49	8T22

Groups of order 48

		Label	ID	Tr ID
GL₂(𝔽₃)	General linear group on 𝔽₃²; = Q₈ ⋊S₃ = Aut(C₃²)	GL(2,3)	48,29	8T23
C₂×S₄	Direct product of C₂ and S₄; = O₃(𝔽₃) = cube/octahedron symmetries	C2xS4	48,48	8T24

Groups of order 56

		Label	ID	Tr ID
F₈	Frobenius group; = C₂³ ⋊C₇ = AGL₁(𝔽₈)	F8	56,11	8T25

Groups of order 64

		Label	ID	Tr ID
D₄⋊₄D₄	3^rd semidirect product of D₄ and D₄ acting via D₄/C₂²=C₂; = Hol(D₄)	D4:4D4	64,134	8T26
C₂≀C₄	Wreath product of C₂ by C₄; = AΣL₁(𝔽₁₆)	C2wrC4	64,32	8T27
				8T28
C₂≀C₂²	Wreath product of C₂ by C₂²; = Hol(C₂×C₄)	C2wrC2^2	64,138	8T29
C₄²⋊C₄	2^nd semidirect product of C₄² and C₄ acting faithfully	C4^2:C4	64,34	8T30
C₂≀C₂²	Wreath product of C₂ by C₂²; = Hol(C₂×C₄)	C2wrC2^2	64,138	8T31

Groups of order 96

		Label	ID	Tr ID
C₂³⋊A₄	2^nd semidirect product of C₂³ and A₄ acting faithfully	C2^3:A4	96,204	8T32
C₂⁴⋊C₆	1^st semidirect product of C₂⁴ and C₆ acting faithfully	C2^4:C6	96,70	8T33
C₂²⋊S₄	The semidirect product of C₂² and S₄ acting via S₄/C₂²=S₃	C2^2:S4	96,227	8T34

Groups of order 128

		Label	ID	Tr ID
D₄≀C₂	Wreath product of D₄ by C₂	D4wrC2	128,928	8T35

Groups of order 168

		Label	ID	Tr ID
AΓL₁(𝔽₈)	Affine semilinear group on 𝔽₈¹; = F₈ ⋊C₃ = Aut(F₈)	AGammaL(1,8)	168,43	8T36
GL₃(𝔽₂)	General linear group on 𝔽₂³; = Aut(C₂³) = L₃(2) = L₂(7); 2^nd non-abelian simple	GL(3,2)	168,42	8T37

Groups of order 192

		Label	ID	Tr ID
C₂≀A₄	Wreath product of C₂ by A₄	C2wrA4	192,201	8T38
C₂³⋊S₄	2^nd semidirect product of C₂³ and S₄ acting faithfully; = Aut(C₂²×C₄)	C2^3:S4	192,1493	8T39
Q₈⋊₂S₄	2^nd semidirect product of Q₈ and S₄ acting via S₄/C₂²=S₃; = Hol(Q₈)	Q8:2S4	192,1494	8T40
C₂⁴⋊D₆	1^st semidirect product of C₂⁴ and D₆ acting faithfully; = Aut(C₂×Q₈)	C2^4:D6	192,955	8T41

Groups of order 288

		Label	ID	Tr ID
A₄≀C₂	Wreath product of A₄ by C₂	A4wrC2	288,1025	8T42

Groups of order 336

		Label	ID	Tr ID
PGL₂(𝔽₇)	Projective linear group on 𝔽₇²; = GL₃(𝔽₂)⋊C₂ = Aut(GL₃(𝔽₂)); almost simple	PGL(2,7)	336,208	8T43

On 9 points

Groups of order 9

		Label	ID	Tr ID
C₉	Cyclic group	C9	9,1	9T1
C₃²	Elementary abelian group of type [3,3]	C3^2	9,2	9T2

Groups of order 18

		Label	ID	Tr ID
D₉	Dihedral group	D9	18,1	9T3
C₃×S₃	Direct product of C₃ and S₃; = U₂(𝔽₂)	C3xS3	18,3	9T4
C₃⋊S₃	The semidirect product of C₃ and S₃ acting via S₃/C₃=C₂	C3:S3	18,4	9T5

Groups of order 27

		Label	ID	Tr ID
3_-¹⁺²	Extraspecial group	ES-(3,1)	27,4	9T6
He₃	Heisenberg group; = C₃² ⋊C₃ = 3₊¹⁺²	He3	27,3	9T7

Groups of order 36

		Label	ID	Tr ID
S₃²	Direct product of S₃ and S₃; = Spin+₄(𝔽₂) = Hol(S₃)	S3^2	36,10	9T8
C₃²⋊C₄	The semidirect product of C₃² and C₄ acting faithfully	C3^2:C4	36,9	9T9

Groups of order 54

		Label	ID	Tr ID
C₉⋊C₆	The semidirect product of C₉ and C₆ acting faithfully; = Aut(D₉) = Hol(C₉)	C9:C6	54,6	9T10
C₃²⋊C₆	The semidirect product of C₃² and C₆ acting faithfully	C3^2:C6	54,5	9T11
He₃⋊C₂	2^nd semidirect product of He₃ and C₂ acting faithfully; = Aut(3_-¹⁺²)	He3:C2	54,8	9T12
C₃²⋊C₆	The semidirect product of C₃² and C₆ acting faithfully	C3^2:C6	54,5	9T13

Groups of order 72

		Label	ID	Tr ID
PSU₃(𝔽₂)	Projective special unitary group on 𝔽₂³; = C₃² ⋊Q₈ = M₉	PSU(3,2)	72,41	9T14
F₉	Frobenius group; = C₃² ⋊C₈ = AGL₁(𝔽₉)	F9	72,39	9T15
S₃≀C₂	Wreath product of S₃ by C₂; = SO+₄(𝔽₂)	S3wrC2	72,40	9T16

Groups of order 81

		Label	ID	Tr ID
C₃≀C₃	Wreath product of C₃ by C₃; = AΣL₁(𝔽₂₇)	C3wrC3	81,7	9T17

Groups of order 108

		Label	ID	Tr ID
C₃²⋊D₆	The semidirect product of C₃² and D₆ acting faithfully	C3^2:D6	108,17	9T18

Groups of order 144

		Label	ID	Tr ID
AΓL₁(𝔽₉)	Affine semilinear group on 𝔽₉¹; = F₉ ⋊C₂ = Aut(C₃²⋊C₄)	AGammaL(1,9)	144,182	9T19

Groups of order 162

		Label	ID	Tr ID
C₃≀S₃	Wreath product of C₃ by S₃	C3wrS3	162,10	9T20
C₃³⋊S₃	2^nd semidirect product of C₃³ and S₃ acting faithfully	C3^3:S3	162,19	9T21
C₃³⋊C₆	1^st semidirect product of C₃³ and C₆ acting faithfully	C3^3:C6	162,11	9T22

Groups of order 216

		Label	ID	Tr ID
ASL₂(𝔽₃)	Hessian group = Affine special linear group on 𝔽₃²; = PSU₃(𝔽₂)⋊C₃	ASL(2,3)	216,153	9T23

Groups of order 324

		Label	ID	Tr ID
He₃⋊D₆	The semidirect product of He₃ and D₆ acting faithfully	He3:D6	324,39	9T24
C₃³⋊A₄	The semidirect product of C₃³ and A₄ acting faithfully	C3^3:A4	324,160	9T25

Groups of order 432

		Label	ID	Tr ID
AGL₂(𝔽₃)	Affine linear group on 𝔽₃²; = PSU₃(𝔽₂)⋊S₃ = Aut(C₃⋊S₃) = Hol(C₃²)	AGL(2,3)	432,734	9T26

On 10 points

Groups of order 10

		Label	ID	Tr ID
C₁₀	Cyclic group	C10	10,2	10T1
D₅	Dihedral group; = pentagon symmetries	D5	10,1	10T2

Groups of order 20

		Label	ID	Tr ID
D₁₀	Dihedral group; = C₂×D₅	D10	20,4	10T3
F₅	Frobenius group; = C₅ ⋊C₄ = AGL₁(𝔽₅) = Aut(D₅) = Hol(C₅) = Sz(2)	F5	20,3	10T4

Groups of order 40

		Label	ID	Tr ID
C₂×F₅	Direct product of C₂ and F₅; = Aut(D₁₀) = Hol(C₁₀)	C2xF5	40,12	10T5

Groups of order 50

		Label	ID	Tr ID
C₅×D₅	Direct product of C₅ and D₅; = AΣL₁(𝔽₂₅)	C5xD5	50,3	10T6

Groups of order 60

		Label	ID	Tr ID
A₅	Alternating group on 5 letters; = SL₂(𝔽₄) = L₂(5) = L₂(4) = icosahedron/dodecahedron rotations; 1^st non-abelian simple	A5	60,5	10T7

Groups of order 80

		Label	ID	Tr ID
C₂⁴⋊C₅	The semidirect product of C₂⁴ and C₅ acting faithfully	C2^4:C5	80,49	10T8

Groups of order 100

		Label	ID	Tr ID
D₅²	Direct product of D₅ and D₅	D5^2	100,13	10T9
C₅²⋊C₄	4^th semidirect product of C₅² and C₄ acting faithfully	C5^2:C4	100,12	10T10

Groups of order 120

		Label	ID	Tr ID
C₂×A₅	Direct product of C₂ and A₅; = icosahedron/dodecahedron symmetries	C2xA5	120,35	10T11
S₅	Symmetric group on 5 letters; = PGL₂(𝔽₅) = Aut(A₅) = 5-cell symmetries; almost simple	S5	120,34	10T12
				10T13

Groups of order 160

		Label	ID	Tr ID
C₂×C₂⁴⋊C₅	Direct product of C₂ and C₂⁴⋊C₅; = AΣL₁(𝔽₃₂)	C2xC2^4:C5	160,235	10T14
C₂⁴⋊D₅	The semidirect product of C₂⁴ and D₅ acting faithfully	C2^4:D5	160,234	10T15
				10T16

Groups of order 200

		Label	ID	Tr ID
D₅⋊F₅	The semidirect product of D₅ and F₅ acting via F₅/D₅=C₂; = Hol(D₅)	D5:F5	200,42	10T17
C₅²⋊C₈	The semidirect product of C₅² and C₈ acting faithfully	C5^2:C8	200,40	10T18
D₅≀C₂	Wreath product of D₅ by C₂	D5wrC2	200,43	10T19
C₅²⋊Q₈	The semidirect product of C₅² and Q₈ acting faithfully	C5^2:Q8	200,44	10T20
D₅≀C₂	Wreath product of D₅ by C₂	D5wrC2	200,43	10T21

Groups of order 240

		Label	ID	Tr ID
C₂×S₅	Direct product of C₂ and S₅; = O₃(𝔽₅)	C2xS5	240,189	10T22

Groups of order 320

		Label	ID	Tr ID
C₂×C₂⁴⋊D₅	Direct product of C₂ and C₂⁴⋊D₅	C2xC2^4:D5	320,1636	10T23
C₂⁴⋊F₅	The semidirect product of C₂⁴ and F₅ acting faithfully	C2^4:F5	320,1635	10T24
				10T25

Groups of order 360

		Label	ID	Tr ID
A₆	Alternating group on 6 letters; = PSL₂(𝔽₉) = L₂(9); 3^rd non-abelian simple	A6	360,118	10T26

Groups of order 400

		Label	ID	Tr ID
D₅≀C₂⋊C₂	The semidirect product of D₅≀C₂ and C₂ acting faithfully	D5wrC2:C2	400,207	10T27
C₅²⋊M₄(2)	The semidirect product of C₅² and M₄(2) acting faithfully	C5^2:M4(2)	400,206	10T28

On 11 points

Groups of order 11

		Label	ID	Tr ID
C₁₁	Cyclic group	C11	11,1	11T1

Groups of order 22

		Label	ID	Tr ID
D₁₁	Dihedral group	D11	22,1	11T2

Groups of order 55

		Label	ID	Tr ID
C₁₁⋊C₅	The semidirect product of C₁₁ and C₅ acting faithfully	C11:C5	55,1	11T3

Groups of order 110

		Label	ID	Tr ID
F₁₁	Frobenius group; = C₁₁ ⋊C₁₀ = AGL₁(𝔽₁₁) = Aut(D₁₁) = Hol(C₁₁)	F11	110,1	11T4

On 12 points

Groups of order 12

		Label	ID	Tr ID
C₁₂	Cyclic group	C12	12,2	12T1
C₂×C₆	Abelian group of type [2,6]	C2xC6	12,5	12T2
D₆	Dihedral group; = C₂×S₃ = hexagon symmetries	D6	12,4	12T3
A₄	Alternating group on 4 letters; = PSL₂(𝔽₃) = L₂(3) = tetrahedron rotations	A4	12,3	12T4
Dic₃	Dicyclic group; = C₃ ⋊C₄	Dic3	12,1	12T5

Groups of order 24

		Label	ID	Tr ID
C₂×A₄	Direct product of C₂ and A₄; = AΣL₁(𝔽₈)	C2xA4	24,13	12T6
				12T7
S₄	Symmetric group on 4 letters; = PGL₂(𝔽₃) = Aut(Q₈) = Hol(C₂²) = tetrahedron symmetries = cube/octahedron rotations	S4	24,12	12T8
				12T9
C₂²×S₃	Direct product of C₂² and S₃	C2^2xS3	24,14	12T10
C₄×S₃	Direct product of C₄ and S₃	C4xS3	24,5	12T11
D₁₂	Dihedral group	D12	24,6	12T12
C₃⋊D₄	The semidirect product of C₃ and D₄ acting via D₄/C₂²=C₂	C3:D4	24,8	12T13
C₃×D₄	Direct product of C₃ and D₄	C3xD4	24,10	12T14
C₃⋊D₄	The semidirect product of C₃ and D₄ acting via D₄/C₂²=C₂	C3:D4	24,8	12T15

Groups of order 36

		Label	ID	Tr ID
S₃²	Direct product of S₃ and S₃; = Spin+₄(𝔽₂) = Hol(S₃)	S3^2	36,10	12T16
C₃²⋊C₄	The semidirect product of C₃² and C₄ acting faithfully	C3^2:C4	36,9	12T17
S₃×C₆	Direct product of C₆ and S₃	S3xC6	36,12	12T18
C₃×Dic₃	Direct product of C₃ and Dic₃	C3xDic3	36,6	12T19
C₃×A₄	Direct product of C₃ and A₄	C3xA4	36,11	12T20

Groups of order 48

		Label	ID	Tr ID
C₂×S₄	Direct product of C₂ and S₄; = O₃(𝔽₃) = cube/octahedron symmetries	C2xS4	48,48	12T21
				12T22
				12T23
				12T24
C₂²×A₄	Direct product of C₂² and A₄	C2^2xA4	48,49	12T25
				12T26
A₄⋊C₄	The semidirect product of A₄ and C₄ acting via C₄/C₂=C₂; = SL₂(ℤ/4ℤ)	A4:C4	48,30	12T27
S₃×D₄	Direct product of S₃ and D₄; = Aut(D₁₂) = Hol(C₁₂)	S3xD4	48,38	12T28
C₄×A₄	Direct product of C₄ and A₄	C4xA4	48,31	12T29
A₄⋊C₄	The semidirect product of A₄ and C₄ acting via C₄/C₂=C₂; = SL₂(ℤ/4ℤ)	A4:C4	48,30	12T30
C₄²⋊C₃	The semidirect product of C₄² and C₃ acting faithfully	C4^2:C3	48,3	12T31
C₂²⋊A₄	The semidirect product of C₂² and A₄ acting via A₄/C₂²=C₃	C2^2:A4	48,50	12T32

Groups of order 60

		Label	ID	Tr ID
A₅	Alternating group on 5 letters; = SL₂(𝔽₄) = L₂(5) = L₂(4) = icosahedron/dodecahedron rotations; 1^st non-abelian simple	A5	60,5	12T33

Groups of order 72

		Label	ID	Tr ID
S₃≀C₂	Wreath product of S₃ by C₂; = SO+₄(𝔽₂)	S3wrC2	72,40	12T34
				12T35
				12T36
C₂×S₃²	Direct product of C₂, S₃ and S₃	C2xS3^2	72,46	12T37
C₃⋊D₁₂	The semidirect product of C₃ and D₁₂ acting via D₁₂/D₆=C₂	C3:D12	72,23	12T38
C₆.D₆	2^nd non-split extension by C₆ of D₆ acting via D₆/S₃=C₂	C6.D6	72,21	12T39
C₂×C₃²⋊C₄	Direct product of C₂ and C₃²⋊C₄	C2xC3^2:C4	72,45	12T40
				12T41
C₃×C₃⋊D₄	Direct product of C₃ and C₃⋊D₄	C3xC3:D4	72,30	12T42
S₃×A₄	Direct product of S₃ and A₄	S3xA4	72,44	12T43
C₃⋊S₄	The semidirect product of C₃ and S₄ acting via S₄/A₄=C₂	C3:S4	72,43	12T44
C₃×S₄	Direct product of C₃ and S₄	C3xS4	72,42	12T45
F₉	Frobenius group; = C₃² ⋊C₈ = AGL₁(𝔽₉)	F9	72,39	12T46
PSU₃(𝔽₂)	Projective special unitary group on 𝔽₂³; = C₃² ⋊Q₈ = M₉	PSU(3,2)	72,41	12T47

Groups of order 96

		Label	ID	Tr ID
C₂²×S₄	Direct product of C₂² and S₄	C2^2xS4	96,226	12T48
A₄⋊D₄	The semidirect product of A₄ and D₄ acting via D₄/C₂²=C₂; = Aut(C₄²) = GL₂(ℤ/4ℤ)	A4:D4	96,195	12T49
				12T50
D₄×A₄	Direct product of D₄ and A₄	D4xA4	96,197	12T51
A₄⋊D₄	The semidirect product of A₄ and D₄ acting via D₄/C₂²=C₂; = Aut(C₄²) = GL₂(ℤ/4ℤ)	A4:D4	96,195	12T52
C₄×S₄	Direct product of C₄ and S₄	C4xS4	96,186	12T53
C₄⋊S₄	The semidirect product of C₄ and S₄ acting via S₄/A₄=C₂	C4:S4	96,187	12T54
C₂×C₄²⋊C₃	Direct product of C₂ and C₄²⋊C₃	C2xC4^2:C3	96,68	12T55
C₂×C₂²⋊A₄	Direct product of C₂ and C₂²⋊A₄	C2xC2^2:A4	96,229	12T56
C₂³.₃A₄	1^st non-split extension by C₂³ of A₄ acting via A₄/C₂²=C₃	C2^3.3A4	96,3	12T57
C₂⁴⋊C₆	1^st semidirect product of C₂⁴ and C₆ acting faithfully	C2^4:C6	96,70	12T58
				12T59
C₂³.A₄	2^nd non-split extension by C₂³ of A₄ acting faithfully	C2^3.A4	96,72	12T60
				12T61
C₄²⋊S₃	The semidirect product of C₄² and S₃ acting faithfully	C4^2:S3	96,64	12T62
				12T63
				12T64
				12T65
C₂²⋊S₄	The semidirect product of C₂² and S₄ acting via S₄/C₂²=S₃	C2^2:S4	96,227	12T66
				12T67
				12T68
				12T69

Groups of order 108

		Label	ID	Tr ID
C₃×S₃²	Direct product of C₃, S₃ and S₃	C3xS3^2	108,38	12T70
C₃²⋊₄D₆	The semidirect product of C₃² and D₆ acting via D₆/C₃=C₂²	C3^2:4D6	108,40	12T71
C₃³⋊C₄	2^nd semidirect product of C₃³ and C₄ acting faithfully	C3^3:C4	108,37	12T72
C₃×C₃²⋊C₄	Direct product of C₃ and C₃²⋊C₄	C3xC3^2:C4	108,36	12T73

Groups of order 120

		Label	ID	Tr ID
S₅	Symmetric group on 5 letters; = PGL₂(𝔽₅) = Aut(A₅) = 5-cell symmetries; almost simple	S5	120,34	12T74
C₂×A₅	Direct product of C₂ and A₅; = icosahedron/dodecahedron symmetries	C2xA5	120,35	12T75
				12T76

Groups of order 144

		Label	ID	Tr ID
C₂×S₃≀C₂	Direct product of C₂ and S₃≀C₂	C2xS3wrC2	144,186	12T77
				12T78
S₃²⋊C₄	The semidirect product of S₃² and C₄ acting via C₄/C₂=C₂	S3^2:C4	144,115	12T79
				12T80
Dic₃⋊D₆	2^nd semidirect product of Dic₃ and D₆ acting via D₆/S₃=C₂; = Hol(Dic₃)	Dic3:D6	144,154	12T81
C₆²⋊C₄	1^st semidirect product of C₆² and C₄ acting faithfully	C6^2:C4	144,136	12T82
S₃×S₄	Direct product of S₃ and S₄; = Hol(C₂×C₆)	S3xS4	144,183	12T83
AΓL₁(𝔽₉)	Affine semilinear group on 𝔽₉¹; = F₉ ⋊C₂ = Aut(C₃²⋊C₄)	AGammaL(1,9)	144,182	12T84
A₄²	Direct product of A₄ and A₄; = PΩ+₄(𝔽₃)	A4^2	144,184	12T85

Groups of order 192

		Label	ID	Tr ID
D₄×S₄	Direct product of D₄ and S₄	D4xS4	192,1472	12T86
C₂×C₂⁴⋊C₆	Direct product of C₂ and C₂⁴⋊C₆	C2xC2^4:C6	192,1000	12T87
				12T88
C₂×C₂³.A₄	Direct product of C₂ and C₂³.A₄	C2xC2^3.A4	192,1002	12T89
C₂²×C₂²⋊A₄	Direct product of C₂² and C₂²⋊A₄	C2^2xC2^2:A4	192,1540	12T90
C₂⁴.₂A₄	2^nd non-split extension by C₂⁴ of A₄ acting faithfully	C2^4.2A4	192,197	12T91
C₂×C₂³.A₄	Direct product of C₂ and C₂³.A₄	C2xC2^3.A4	192,1002	12T92
C₂⁴.₂A₄	2^nd non-split extension by C₂⁴ of A₄ acting faithfully	C2^4.2A4	192,197	12T93
C₄×C₄²⋊C₃	Direct product of C₄ and C₄²⋊C₃	C4xC4^2:C3	192,188	12T94
C₂×C₄²⋊S₃	Direct product of C₂ and C₄²⋊S₃	C2xC4^2:S3	192,944	12T95
				12T96
				12T97
C₂³.₉S₄	3^rd non-split extension by C₂³ of S₄ acting via S₄/C₂²=S₃	C2^3.9S4	192,182	12T98
C₂⁴⋊C₁₂	1^st semidirect product of C₂⁴ and C₁₂ acting via C₁₂/C₂=C₆	C2^4:C12	192,191	12T99
C₂×C₂²⋊S₄	Direct product of C₂ and C₂²⋊S₄	C2xC2^2:S4	192,1538	12T100
				12T101
C₂⁴⋊₄Dic₃	3^rd semidirect product of C₂⁴ and Dic₃ acting via Dic₃/C₂=S₃	C2^4:4Dic3	192,1495	12T102
C₂×C₂²⋊S₄	Direct product of C₂ and C₂²⋊S₄	C2xC2^2:S4	192,1538	12T103
C₂³⋊₂D₄⋊C₃	The semidirect product of C₂³⋊₂D₄ and C₃ acting faithfully	C2^3:2D4:C3	192,194	12T104
C₂⁴⋊C₁₂	1^st semidirect product of C₂⁴ and C₁₂ acting via C₁₂/C₂=C₆	C2^4:C12	192,191	12T105
C₂×C₂²⋊S₄	Direct product of C₂ and C₂²⋊S₄	C2xC2^2:S4	192,1538	12T106
C₂⁴⋊₄Dic₃	3^rd semidirect product of C₂⁴ and Dic₃ acting via Dic₃/C₂=S₃	C2^4:4Dic3	192,1495	12T107
C₂⁴⋊D₆	1^st semidirect product of C₂⁴ and D₆ acting faithfully; = Aut(C₂×Q₈)	C2^4:D6	192,955	12T108
				12T109
				12T110
				12T111
C₄²⋊D₆	The semidirect product of C₄² and D₆ acting faithfully	C4^2:D6	192,956	12T112
				12T113
				12T114
				12T115

Groups of order 216

		Label	ID	Tr ID
C₃³⋊D₄	2^nd semidirect product of C₃³ and D₄ acting faithfully	C3^3:D4	216,158	12T116
S₃³	Direct product of S₃, S₃ and S₃; = Hol(C₃×S₃)	S3^3	216,162	12T117
C₃²⋊₂D₁₂	The semidirect product of C₃² and D₁₂ acting via D₁₂/C₃=D₄	C3^2:2D12	216,159	12T118
S₃×C₃²⋊C₄	Direct product of S₃ and C₃²⋊C₄	S3xC3^2:C4	216,156	12T119
C₃³⋊D₄	2^nd semidirect product of C₃³ and D₄ acting faithfully	C3^3:D4	216,158	12T120
C₃×S₃≀C₂	Direct product of C₃ and S₃≀C₂	C3xS3wrC2	216,157	12T121
ASL₂(𝔽₃)	Hessian group = Affine special linear group on 𝔽₃²; = PSU₃(𝔽₂)⋊C₃	ASL(2,3)	216,153	12T122

Groups of order 240

		Label	ID	Tr ID
C₂×S₅	Direct product of C₂ and S₅; = O₃(𝔽₅)	C2xS5	240,189	12T123
A₅⋊C₄	The semidirect product of A₅ and C₄ acting via C₄/C₂=C₂	A5:C4	240,91	12T124

Groups of order 288

		Label	ID	Tr ID
D₆≀C₂	Wreath product of D₆ by C₂	D6wrC2	288,889	12T125
A₄≀C₂	Wreath product of A₄ by C₂	A4wrC2	288,1025	12T126
PSO₄⁺ (𝔽₃)	Projective special orthogonal group of + type on 𝔽₃⁴; = A₄ ⋊S₄ = Hol(A₄)	PSO+(4,3)	288,1026	12T127
A₄≀C₂	Wreath product of A₄ by C₂	A4wrC2	288,1025	12T128
				12T129

Groups of order 324

		Label	ID	Tr ID
C₃×C₃²⋊₄D₆	Direct product of C₃ and C₃²⋊₄D₆	C3xC3^2:4D6	324,167	12T130
C₃×C₃³⋊C₄	Direct product of C₃ and C₃³⋊C₄; = AΣL₁(𝔽₈₁)	C3xC3^3:C4	324,162	12T131
C₃³⋊A₄	The semidirect product of C₃³ and A₄ acting faithfully	C3^3:A4	324,160	12T132
				12T133

Groups of order 432

		Label	ID	Tr ID
S₃×S₃≀C₂	Direct product of S₃ and S₃≀C₂	S3xS3wrC2	432,741	12T156
AGL₂(𝔽₃)	Affine linear group on 𝔽₃²; = PSU₃(𝔽₂)⋊S₃ = Aut(C₃⋊S₃) = Hol(C₃²)	AGL(2,3)	432,734	12T157

On 13 points

Groups of order 13

		Label	ID	Tr ID
C₁₃	Cyclic group	C13	13,1	13T1

Groups of order 26

		Label	ID	Tr ID
D₁₃	Dihedral group	D13	26,1	13T2

Groups of order 39

		Label	ID	Tr ID
C₁₃⋊C₃	The semidirect product of C₁₃ and C₃ acting faithfully	C13:C3	39,1	13T3

Groups of order 52

		Label	ID	Tr ID
C₁₃⋊C₄	The semidirect product of C₁₃ and C₄ acting faithfully	C13:C4	52,3	13T4

Groups of order 78

		Label	ID	Tr ID
C₁₃⋊C₆	The semidirect product of C₁₃ and C₆ acting faithfully	C13:C6	78,1	13T5

Groups of order 156

		Label	ID	Tr ID
F₁₃	Frobenius group; = C₁₃ ⋊C₁₂ = AGL₁(𝔽₁₃) = Aut(D₁₃) = Hol(C₁₃)	F13	156,7	13T6

On 14 points

Groups of order 14

		Label	ID	Tr ID
C₁₄	Cyclic group	C14	14,2	14T1
D₇	Dihedral group	D7	14,1	14T2

Groups of order 28

		Label	ID	Tr ID
D₁₄	Dihedral group; = C₂×D₇	D14	28,3	14T3

Groups of order 42

		Label	ID	Tr ID
F₇	Frobenius group; = C₇ ⋊C₆ = AGL₁(𝔽₇) = Aut(D₇) = Hol(C₇)	F7	42,1	14T4
C₂×C₇⋊C₃	Direct product of C₂ and C₇⋊C₃	C2xC7:C3	42,2	14T5

Groups of order 56

		Label	ID	Tr ID
F₈	Frobenius group; = C₂³ ⋊C₇ = AGL₁(𝔽₈)	F8	56,11	14T6

Groups of order 84

		Label	ID	Tr ID
C₂×F₇	Direct product of C₂ and F₇; = Aut(D₁₄) = Hol(C₁₄)	C2xF7	84,7	14T7

Groups of order 98

		Label	ID	Tr ID
C₇×D₇	Direct product of C₇ and D₇; = AΣL₁(𝔽₄₉)	C7xD7	98,3	14T8

Groups of order 112

		Label	ID	Tr ID
C₂×F₈	Direct product of C₂ and F₈	C2xF8	112,41	14T9

Groups of order 168

		Label	ID	Tr ID
GL₃(𝔽₂)	General linear group on 𝔽₂³; = Aut(C₂³) = L₃(2) = L₂(7); 2^nd non-abelian simple	GL(3,2)	168,42	14T10
AΓL₁(𝔽₈)	Affine semilinear group on 𝔽₈¹; = F₈ ⋊C₃ = Aut(F₈)	AGammaL(1,8)	168,43	14T11

Groups of order 196

		Label	ID	Tr ID
C₇²⋊C₄	The semidirect product of C₇² and C₄ acting faithfully	C7^2:C4	196,8	14T12
D₇²	Direct product of D₇ and D₇	D7^2	196,9	14T13

Groups of order 294

		Label	ID	Tr ID
C₇⋊₄F₇	2^nd semidirect product of C₇ and F₇ acting via F₇/D₇=C₃	C7:4F7	294,12	14T14
C₇²⋊S₃	The semidirect product of C₇² and S₃ acting faithfully	C7^2:S3	294,7	14T15

Groups of order 336

		Label	ID	Tr ID
PGL₂(𝔽₇)	Projective linear group on 𝔽₇²; = GL₃(𝔽₂)⋊C₂ = Aut(GL₃(𝔽₂)); almost simple	PGL(2,7)	336,208	14T16
C₂×GL₃(𝔽₂)	Direct product of C₂ and GL₃(𝔽₂)	C2xGL(3,2)	336,209	14T17
C₂×AΓL₁(𝔽₈)	Direct product of C₂ and AΓL₁(𝔽₈)	C2xAGammaL(1,8)	336,210	14T18
C₂×GL₃(𝔽₂)	Direct product of C₂ and GL₃(𝔽₂)	C2xGL(3,2)	336,209	14T19

Groups of order 392

		Label	ID	Tr ID
D₇≀C₂	Wreath product of D₇ by C₂	D7wrC2	392,37	14T20

Groups of order 448

		Label	ID	Tr ID
C₂³⋊F₈	2^nd semidirect product of C₂³ and F₈ acting via F₈/C₂³=C₇	C2^3:F8	448,1394	14T21

On 15 points

Groups of order 15

		Label	ID	Tr ID
C₁₅	Cyclic group	C15	15,1	15T1

Groups of order 30

		Label	ID	Tr ID
D₁₅	Dihedral group	D15	30,3	15T2
C₃×D₅	Direct product of C₃ and D₅	C3xD5	30,2	15T3
C₅×S₃	Direct product of C₅ and S₃	C5xS3	30,1	15T4

Groups of order 60

		Label	ID	Tr ID
A₅	Alternating group on 5 letters; = SL₂(𝔽₄) = L₂(5) = L₂(4) = icosahedron/dodecahedron rotations; 1^st non-abelian simple	A5	60,5	15T5
C₃⋊F₅	The semidirect product of C₃ and F₅ acting via F₅/D₅=C₂	C3:F5	60,7	15T6
S₃×D₅	Direct product of S₃ and D₅	S3xD5	60,8	15T7
C₃×F₅	Direct product of C₃ and F₅	C3xF5	60,6	15T8

Groups of order 75

		Label	ID	Tr ID
C₅²⋊C₃	The semidirect product of C₅² and C₃ acting faithfully	C5^2:C3	75,2	15T9

Groups of order 120

		Label	ID	Tr ID
S₅	Symmetric group on 5 letters; = PGL₂(𝔽₅) = Aut(A₅) = 5-cell symmetries; almost simple	S5	120,34	15T10
S₃×F₅	Direct product of S₃ and F₅; = Aut(D₁₅) = Hol(C₁₅)	S3xF5	120,36	15T11

Groups of order 150

		Label	ID	Tr ID
C₅²⋊C₆	The semidirect product of C₅² and C₆ acting faithfully	C5^2:C6	150,6	15T12
C₅²⋊S₃	The semidirect product of C₅² and S₃ acting faithfully	C5^2:S3	150,5	15T13
				15T14

Groups of order 180

		Label	ID	Tr ID
C₃×A₅	Direct product of C₃ and A₅; = GL₂(𝔽₄)	C3xA5	180,19	15T15
				15T16

Groups of order 300

		Label	ID	Tr ID
C₅²⋊Dic₃	The semidirect product of C₅² and Dic₃ acting faithfully	C5^2:Dic3	300,23	15T17
C₅²⋊D₆	The semidirect product of C₅² and D₆ acting faithfully	C5^2:D6	300,25	15T18
C₅²⋊C₁₂	The semidirect product of C₅² and C₁₂ acting faithfully	C5^2:C12	300,24	15T19

Groups of order 360

		Label	ID	Tr ID
A₆	Alternating group on 6 letters; = PSL₂(𝔽₉) = L₂(9); 3^rd non-abelian simple	A6	360,118	15T20
ΓL₂(𝔽₄)	Semilinear group on 𝔽₄²; = C₃ ⋊S₅	GammaL(2,4)	360,120	15T21
				15T22
S₃×A₅	Direct product of S₃ and A₅	S3xA5	360,121	15T23
C₃×S₅	Direct product of C₃ and S₅	C3xS5	360,119	15T24

Groups of order 375

		Label	ID	Tr ID
C₅×C₅²⋊C₃	Direct product of C₅ and C₅²⋊C₃; = AΣL₁(𝔽₁₂₅)	C5xC5^2:C3	375,6	15T25

Groups of order 405

		Label	ID	Tr ID
C₃⁴⋊C₅	The semidirect product of C₃⁴ and C₅ acting faithfully	C3^4:C5	405,15	15T26

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