Group​Names [Beta]

Finite groups of order ≤500, group names, extensions, presentations, properties and character tables.

Order  ≤60≤120≤250≤500

Orders with >300 groups of order n
n = 128, 192, 256, 288, 320, 384, 432, 448, 480.

[non]​abelian, [non]​soluble, super​soluble, [non]​monomial, Z-groups, A-groups, metacyclic, metabelian, p-groups, elementary, hyper​elementary, linear, perfect, simple, almost simple, quasisimple, rational groups.

Groups of order 1

dρLabelID
C1Trivial group11+C11,1

Groups of order 2

dρLabelID
C2Cyclic group21+C22,1

Groups of order 3

dρLabelID
C3Cyclic group; = A3 = triangle rotations31C33,1

Groups of order 4

dρLabelID
C4Cyclic group; = square rotations41C44,1
C22Klein 4-group V4 = elementary abelian group of type [2,2]; = rectangle symmetries4C2^24,2

Groups of order 5

dρLabelID
C5Cyclic group; = pentagon rotations51C55,1

Groups of order 6

dρLabelID
C6Cyclic group; = hexagon rotations61C66,2
S3Symmetric group on 3 letters; = D3 = GL2(𝔽2) = triangle symmetries = 1st non-abelian group32+S36,1

Groups of order 7

dρLabelID
C7Cyclic group71C77,1

Groups of order 8

dρLabelID
C8Cyclic group81C88,1
D4Dihedral group; = He2 = AΣL1(𝔽4) = 2+ 1+2 = square symmetries42+D48,3
Q8Quaternion group; = C4.C2 = Dic2 = 2- 1+282-Q88,4
C23Elementary abelian group of type [2,2,2]8C2^38,5
C2×C4Abelian group of type [2,4]8C2xC48,2

Groups of order 9

dρLabelID
C9Cyclic group91C99,1
C32Elementary abelian group of type [3,3]9C3^29,2

Groups of order 10

dρLabelID
C10Cyclic group101C1010,2
D5Dihedral group; = pentagon symmetries52+D510,1

Groups of order 11

dρLabelID
C11Cyclic group111C1111,1

Groups of order 12

dρLabelID
C12Cyclic group121C1212,2
A4Alternating group on 4 letters; = PSL2(𝔽3) = L2(3) = tetrahedron rotations43+A412,3
D6Dihedral group; = C2×S3 = hexagon symmetries62+D612,4
Dic3Dicyclic group; = C3C4122-Dic312,1
C2×C6Abelian group of type [2,6]12C2xC612,5

Groups of order 13

dρLabelID
C13Cyclic group131C1313,1

Groups of order 14

dρLabelID
C14Cyclic group141C1414,2
D7Dihedral group72+D714,1

Groups of order 15

dρLabelID
C15Cyclic group151C1515,1

Groups of order 16

dρLabelID
C16Cyclic group161C1616,1
D8Dihedral group82+D816,7
Q16Generalised quaternion group; = C8.C2 = Dic4162-Q1616,9
SD16Semidihedral group; = Q8C2 = QD1682SD1616,8
M4(2)Modular maximal-cyclic group; = C83C282M4(2)16,6
C4○D4Pauli group = central product of C4 and D482C4oD416,13
C22⋊C4The semidirect product of C22 and C4 acting via C4/C2=C28C2^2:C416,3
C4⋊C4The semidirect product of C4 and C4 acting via C4/C2=C216C4:C416,4
C42Abelian group of type [4,4]16C4^216,2
C24Elementary abelian group of type [2,2,2,2]16C2^416,14
C2×C8Abelian group of type [2,8]16C2xC816,5
C22×C4Abelian group of type [2,2,4]16C2^2xC416,10
C2×D4Direct product of C2 and D48C2xD416,11
C2×Q8Direct product of C2 and Q816C2xQ816,12

Groups of order 17

dρLabelID
C17Cyclic group171C1717,1

Groups of order 18

dρLabelID
C18Cyclic group181C1818,2
D9Dihedral group92+D918,1
C3⋊S3The semidirect product of C3 and S3 acting via S3/C3=C29C3:S318,4
C3×C6Abelian group of type [3,6]18C3xC618,5
C3×S3Direct product of C3 and S3; = U2(𝔽2)62C3xS318,3

Groups of order 19

dρLabelID
C19Cyclic group191C1919,1

Groups of order 20

dρLabelID
C20Cyclic group201C2020,2
D10Dihedral group; = C2×D5102+D1020,4
F5Frobenius group; = C5C4 = AGL1(𝔽5) = Aut(D5) = Hol(C5) = Sz(2)54+F520,3
Dic5Dicyclic group; = C52C4202-Dic520,1
C2×C10Abelian group of type [2,10]20C2xC1020,5

Groups of order 21

dρLabelID
C21Cyclic group211C2121,2
C7⋊C3The semidirect product of C7 and C3 acting faithfully73C7:C321,1

Groups of order 22

dρLabelID
C22Cyclic group221C2222,2
D11Dihedral group112+D1122,1

Groups of order 23

dρLabelID
C23Cyclic group231C2323,1

Groups of order 24

dρLabelID
C24Cyclic group241C2424,2
S4Symmetric group on 4 letters; = PGL2(𝔽3) = Aut(Q8) = Hol(C22) = tetrahedron symmetries = cube/octahedron rotations43+S424,12
D12Dihedral group122+D1224,6
Dic6Dicyclic group; = C3Q8242-Dic624,4
Q8⋊C3The semidirect product of Q8 and C3 acting faithfully; = SL2(𝔽3) = 2T = <2,3,3> = 1st non-monomial group82-Q8:C324,3
C3⋊D4The semidirect product of C3 and D4 acting via D4/C22=C2122C3:D424,8
C3⋊C8The semidirect product of C3 and C8 acting via C8/C4=C2242C3:C824,1
C2×C12Abelian group of type [2,12]24C2xC1224,9
C22×C6Abelian group of type [2,2,6]24C2^2xC624,15
C2×A4Direct product of C2 and A4; = AΣL1(𝔽8)63+C2xA424,13
C4×S3Direct product of C4 and S3122C4xS324,5
C3×D4Direct product of C3 and D4122C3xD424,10
C22×S3Direct product of C22 and S312C2^2xS324,14
C3×Q8Direct product of C3 and Q8242C3xQ824,11
C2×Dic3Direct product of C2 and Dic324C2xDic324,7

Groups of order 25

dρLabelID
C25Cyclic group251C2525,1
C52Elementary abelian group of type [5,5]25C5^225,2

Groups of order 26

dρLabelID
C26Cyclic group261C2626,2
D13Dihedral group132+D1326,1

Groups of order 27

dρLabelID
C27Cyclic group271C2727,1
He3Heisenberg group; = C32C3 = 3+ 1+293He327,3
3- 1+2Extraspecial group93ES-(3,1)27,4
C33Elementary abelian group of type [3,3,3]27C3^327,5
C3×C9Abelian group of type [3,9]27C3xC927,2

Groups of order 28

dρLabelID
C28Cyclic group281C2828,2
D14Dihedral group; = C2×D7142+D1428,3
Dic7Dicyclic group; = C7C4282-Dic728,1
C2×C14Abelian group of type [2,14]28C2xC1428,4

Groups of order 29

dρLabelID
C29Cyclic group291C2929,1

Groups of order 30

dρLabelID
C30Cyclic group301C3030,4
D15Dihedral group152+D1530,3
C5×S3Direct product of C5 and S3152C5xS330,1
C3×D5Direct product of C3 and D5152C3xD530,2

Groups of order 31

dρLabelID
C31Cyclic group311C3131,1

Groups of order 32

dρLabelID
C32Cyclic group321C3232,1
D16Dihedral group162+D1632,18
Q32Generalised quaternion group; = C16.C2 = Dic8322-Q3232,20
2+ 1+4Extraspecial group; = D4D484+ES+(2,2)32,49
SD32Semidihedral group; = C162C2 = QD32162SD3232,19
2- 1+4Gamma matrices = Extraspecial group; = D4Q8164-ES-(2,2)32,50
M5(2)Modular maximal-cyclic group; = C163C2162M5(2)32,17
C4≀C2Wreath product of C4 by C282C4wrC232,11
C22≀C2Wreath product of C22 by C28C2^2wrC232,27
C8○D4Central product of C8 and D4162C8oD432,38
C4○D8Central product of C4 and D8162C4oD832,42
C23⋊C4The semidirect product of C23 and C4 acting faithfully84+C2^3:C432,6
C8⋊C22The semidirect product of C8 and C22 acting faithfully; = Aut(D8) = Hol(C8)84+C8:C2^232,43
C4⋊D4The semidirect product of C4 and D4 acting via D4/C22=C216C4:D432,28
C41D4The semidirect product of C4 and D4 acting via D4/C4=C216C4:1D432,34
C22⋊C8The semidirect product of C22 and C8 acting via C8/C4=C216C2^2:C832,5
C22⋊Q8The semidirect product of C22 and Q8 acting via Q8/C4=C216C2^2:Q832,29
D4⋊C41st semidirect product of D4 and C4 acting via C4/C2=C216D4:C432,9
C42⋊C21st semidirect product of C42 and C2 acting faithfully16C4^2:C232,24
C422C22nd semidirect product of C42 and C2 acting faithfully16C4^2:2C232,33
C4⋊C8The semidirect product of C4 and C8 acting via C8/C4=C232C4:C832,12
C4⋊Q8The semidirect product of C4 and Q8 acting via Q8/C4=C232C4:Q832,35
C8⋊C43rd semidirect product of C8 and C4 acting via C4/C2=C232C8:C432,4
Q8⋊C41st semidirect product of Q8 and C4 acting via C4/C2=C232Q8:C432,10
C4.D41st non-split extension by C4 of D4 acting via D4/C22=C284+C4.D432,7
C8.C41st non-split extension by C8 of C4 acting via C4/C2=C2162C8.C432,15
C4.4D44th non-split extension by C4 of D4 acting via D4/C4=C216C4.4D432,31
C8.C22The non-split extension by C8 of C22 acting faithfully164-C8.C2^232,44
C4.10D42nd non-split extension by C4 of D4 acting via D4/C22=C2164-C4.10D432,8
C22.D43rd non-split extension by C22 of D4 acting via D4/C22=C216C2^2.D432,30
C2.D82nd central extension by C2 of D832C2.D832,14
C4.Q81st non-split extension by C4 of Q8 acting via Q8/C4=C232C4.Q832,13
C2.C421st central stem extension by C2 of C4232C2.C4^232,2
C42.C24th non-split extension by C42 of C2 acting faithfully32C4^2.C232,32
C25Elementary abelian group of type [2,2,2,2,2]32C2^532,51
C4×C8Abelian group of type [4,8]32C4xC832,3
C2×C16Abelian group of type [2,16]32C2xC1632,16
C2×C42Abelian group of type [2,4,4]32C2xC4^232,21
C22×C8Abelian group of type [2,2,8]32C2^2xC832,36
C23×C4Abelian group of type [2,2,2,4]32C2^3xC432,45
C4×D4Direct product of C4 and D416C4xD432,25
C2×D8Direct product of C2 and D816C2xD832,39
C2×SD16Direct product of C2 and SD1616C2xSD1632,40
C22×D4Direct product of C22 and D416C2^2xD432,46
C2×M4(2)Direct product of C2 and M4(2)16C2xM4(2)32,37
C4×Q8Direct product of C4 and Q832C4xQ832,26
C2×Q16Direct product of C2 and Q1632C2xQ1632,41
C22×Q8Direct product of C22 and Q832C2^2xQ832,47
C2×C4○D4Direct product of C2 and C4○D416C2xC4oD432,48
C2×C22⋊C4Direct product of C2 and C22⋊C416C2xC2^2:C432,22
C2×C4⋊C4Direct product of C2 and C4⋊C432C2xC4:C432,23

Groups of order 33

dρLabelID
C33Cyclic group331C3333,1

Groups of order 34

dρLabelID
C34Cyclic group341C3434,2
D17Dihedral group172+D1734,1

Groups of order 35

dρLabelID
C35Cyclic group351C3535,1

Groups of order 36

dρLabelID
C36Cyclic group361C3636,2
D18Dihedral group; = C2×D9182+D1836,4
Dic9Dicyclic group; = C9C4362-Dic936,1
C32⋊C4The semidirect product of C32 and C4 acting faithfully64+C3^2:C436,9
C3⋊Dic3The semidirect product of C3 and Dic3 acting via Dic3/C6=C236C3:Dic336,7
C3.A4The central extension by C3 of A4183C3.A436,3
C62Abelian group of type [6,6]36C6^236,14
C2×C18Abelian group of type [2,18]36C2xC1836,5
C3×C12Abelian group of type [3,12]36C3xC1236,8
S32Direct product of S3 and S3; = Spin+4(𝔽2) = Hol(S3)64+S3^236,10
S3×C6Direct product of C6 and S3122S3xC636,12
C3×A4Direct product of C3 and A4123C3xA436,11
C3×Dic3Direct product of C3 and Dic3122C3xDic336,6
C2×C3⋊S3Direct product of C2 and C3⋊S318C2xC3:S336,13

Groups of order 37

dρLabelID
C37Cyclic group371C3737,1

Groups of order 38

dρLabelID
C38Cyclic group381C3838,2
D19Dihedral group192+D1938,1

Groups of order 39

dρLabelID
C39Cyclic group391C3939,2
C13⋊C3The semidirect product of C13 and C3 acting faithfully133C13:C339,1

Groups of order 40

dρLabelID
C40Cyclic group401C4040,2
D20Dihedral group202+D2040,6
Dic10Dicyclic group; = C5Q8402-Dic1040,4
C5⋊D4The semidirect product of C5 and D4 acting via D4/C22=C2202C5:D440,8
C5⋊C8The semidirect product of C5 and C8 acting via C8/C2=C4404-C5:C840,3
C52C8The semidirect product of C5 and C8 acting via C8/C4=C2402C5:2C840,1
C2×C20Abelian group of type [2,20]40C2xC2040,9
C22×C10Abelian group of type [2,2,10]40C2^2xC1040,14
C2×F5Direct product of C2 and F5; = Aut(D10) = Hol(C10)104+C2xF540,12
C4×D5Direct product of C4 and D5202C4xD540,5
C5×D4Direct product of C5 and D4202C5xD440,10
C22×D5Direct product of C22 and D520C2^2xD540,13
C5×Q8Direct product of C5 and Q8402C5xQ840,11
C2×Dic5Direct product of C2 and Dic540C2xDic540,7

Groups of order 41

dρLabelID
C41Cyclic group411C4141,1

Groups of order 42

dρLabelID
C42Cyclic group421C4242,6
D21Dihedral group212+D2142,5
F7Frobenius group; = C7C6 = AGL1(𝔽7) = Aut(D7) = Hol(C7)76+F742,1
S3×C7Direct product of C7 and S3212S3xC742,3
C3×D7Direct product of C3 and D7212C3xD742,4
C2×C7⋊C3Direct product of C2 and C7⋊C3143C2xC7:C342,2

Groups of order 43

dρLabelID
C43Cyclic group431C4343,1

Groups of order 44

dρLabelID
C44Cyclic group441C4444,2
D22Dihedral group; = C2×D11222+D2244,3
Dic11Dicyclic group; = C11C4442-Dic1144,1
C2×C22Abelian group of type [2,22]44C2xC2244,4

Groups of order 45

dρLabelID
C45Cyclic group451C4545,1
C3×C15Abelian group of type [3,15]45C3xC1545,2

Groups of order 46

dρLabelID
C46Cyclic group461C4646,2
D23Dihedral group232+D2346,1

Groups of order 47

dρLabelID
C47Cyclic group471C4747,1

Groups of order 48

dρLabelID
C48Cyclic group481C4848,2
D24Dihedral group242+D2448,7
Dic12Dicyclic group; = C31Q16482-Dic1248,8
C4○D12Central product of C4 and D12242C4oD1248,37
Q8⋊S3The semidirect product of Q8 and S3 acting faithfully; = GL2(𝔽3) = Aut(C32) = 2O = <2,3,4>82Q8:S348,29
A4⋊C4The semidirect product of A4 and C4 acting via C4/C2=C2; = SL2(ℤ/4ℤ)123A4:C448,30
C42⋊C3The semidirect product of C42 and C3 acting faithfully123C4^2:C348,3
C22⋊A4The semidirect product of C22 and A4 acting via A4/C22=C312C2^2:A448,50
D6⋊C4The semidirect product of D6 and C4 acting via C4/C2=C224D6:C448,14
D4⋊S3The semidirect product of D4 and S3 acting via S3/C3=C2244+D4:S348,15
C8⋊S33rd semidirect product of C8 and S3 acting via S3/C3=C2242C8:S348,5
C24⋊C22nd semidirect product of C24 and C2 acting faithfully242C24:C248,6
D42S3The semidirect product of D4 and S3 acting through Inn(D4)244-D4:2S348,39
Q82S3The semidirect product of Q8 and S3 acting via S3/C3=C2244+Q8:2S348,17
Q83S3The semidirect product of Q8 and S3 acting through Inn(Q8)244+Q8:3S348,41
C3⋊C16The semidirect product of C3 and C16 acting via C16/C8=C2482C3:C1648,1
C4⋊Dic3The semidirect product of C4 and Dic3 acting via Dic3/C6=C248C4:Dic348,13
C3⋊Q16The semidirect product of C3 and Q16 acting via Q16/Q8=C2484-C3:Q1648,18
Dic3⋊C4The semidirect product of Dic3 and C4 acting via C4/C2=C248Dic3:C448,12
C4.A4The central extension by C4 of A4162C4.A448,33
Q8.S3The non-split extension by Q8 of S3 acting faithfully; = CSU2(𝔽3)162-Q8.S348,28
D4.S3The non-split extension by D4 of S3 acting via S3/C3=C2244-D4.S348,16
C4.Dic3The non-split extension by C4 of Dic3 acting via Dic3/C6=C2242C4.Dic348,10
C6.D47th non-split extension by C6 of D4 acting via D4/C22=C224C6.D448,19
C4×C12Abelian group of type [4,12]48C4xC1248,20
C2×C24Abelian group of type [2,24]48C2xC2448,23
C23×C6Abelian group of type [2,2,2,6]48C2^3xC648,52
C22×C12Abelian group of type [2,2,12]48C2^2xC1248,44
C2×S4Direct product of C2 and S4; = O3(𝔽3) = cube/octahedron symmetries63+C2xS448,48
C4×A4Direct product of C4 and A4123C4xA448,31
S3×D4Direct product of S3 and D4; = Aut(D12) = Hol(C12)124+S3xD448,38
C22×A4Direct product of C22 and A412C2^2xA448,49
S3×C8Direct product of C8 and S3242S3xC848,4
C3×D8Direct product of C3 and D8242C3xD848,25
C6×D4Direct product of C6 and D424C6xD448,45
S3×Q8Direct product of S3 and Q8244-S3xQ848,40
C2×D12Direct product of C2 and D1224C2xD1248,36
S3×C23Direct product of C23 and S324S3xC2^348,51
C3×SD16Direct product of C3 and SD16242C3xSD1648,26
C3×M4(2)Direct product of C3 and M4(2)242C3xM4(2)48,24
C6×Q8Direct product of C6 and Q848C6xQ848,46
C3×Q16Direct product of C3 and Q16482C3xQ1648,27
C4×Dic3Direct product of C4 and Dic348C4xDic348,11
C2×Dic6Direct product of C2 and Dic648C2xDic648,34
C22×Dic3Direct product of C22 and Dic348C2^2xDic348,42
C2×Q8⋊C3Direct product of C2 and Q8⋊C316C2xQ8:C348,32
S3×C2×C4Direct product of C2×C4 and S324S3xC2xC448,35
C2×C3⋊D4Direct product of C2 and C3⋊D424C2xC3:D448,43
C3×C4○D4Direct product of C3 and C4○D4242C3xC4oD448,47
C3×C22⋊C4Direct product of C3 and C22⋊C424C3xC2^2:C448,21
C2×C3⋊C8Direct product of C2 and C3⋊C848C2xC3:C848,9
C3×C4⋊C4Direct product of C3 and C4⋊C448C3xC4:C448,22

Groups of order 49

dρLabelID
C49Cyclic group491C4949,1
C72Elementary abelian group of type [7,7]49C7^249,2

Groups of order 50

dρLabelID
C50Cyclic group501C5050,2
D25Dihedral group252+D2550,1
C5⋊D5The semidirect product of C5 and D5 acting via D5/C5=C225C5:D550,4
C5×C10Abelian group of type [5,10]50C5xC1050,5
C5×D5Direct product of C5 and D5; = AΣL1(𝔽25)102C5xD550,3

Groups of order 51

dρLabelID
C51Cyclic group511C5151,1

Groups of order 52

dρLabelID
C52Cyclic group521C5252,2
D26Dihedral group; = C2×D13262+D2652,4
Dic13Dicyclic group; = C132C4522-Dic1352,1
C13⋊C4The semidirect product of C13 and C4 acting faithfully134+C13:C452,3
C2×C26Abelian group of type [2,26]52C2xC2652,5

Groups of order 53

dρLabelID
C53Cyclic group531C5353,1

Groups of order 54

dρLabelID
C54Cyclic group541C5454,2
D27Dihedral group272+D2754,1
C9⋊C6The semidirect product of C9 and C6 acting faithfully; = Aut(D9) = Hol(C9)96+C9:C654,6
C32⋊C6The semidirect product of C32 and C6 acting faithfully96+C3^2:C654,5
He3⋊C22nd semidirect product of He3 and C2 acting faithfully; = Aut(3- 1+2)93He3:C254,8
C9⋊S3The semidirect product of C9 and S3 acting via S3/C3=C227C9:S354,7
C33⋊C23rd semidirect product of C33 and C2 acting faithfully27C3^3:C254,14
C3×C18Abelian group of type [3,18]54C3xC1854,9
C32×C6Abelian group of type [3,3,6]54C3^2xC654,15
S3×C9Direct product of C9 and S3182S3xC954,4
C3×D9Direct product of C3 and D9182C3xD954,3
C2×He3Direct product of C2 and He3183C2xHe354,10
S3×C32Direct product of C32 and S318S3xC3^254,12
C2×3- 1+2Direct product of C2 and 3- 1+2183C2xES-(3,1)54,11
C3×C3⋊S3Direct product of C3 and C3⋊S318C3xC3:S354,13

Groups of order 55

dρLabelID
C55Cyclic group551C5555,2
C11⋊C5The semidirect product of C11 and C5 acting faithfully115C11:C555,1

Groups of order 56

dρLabelID
C56Cyclic group561C5656,2
D28Dihedral group282+D2856,5
F8Frobenius group; = C23C7 = AGL1(𝔽8)87+F856,11
Dic14Dicyclic group; = C7Q8562-Dic1456,3
C7⋊D4The semidirect product of C7 and D4 acting via D4/C22=C2282C7:D456,7
C7⋊C8The semidirect product of C7 and C8 acting via C8/C4=C2562C7:C856,1
C2×C28Abelian group of type [2,28]56C2xC2856,8
C22×C14Abelian group of type [2,2,14]56C2^2xC1456,13
C4×D7Direct product of C4 and D7282C4xD756,4
C7×D4Direct product of C7 and D4282C7xD456,9
C22×D7Direct product of C22 and D728C2^2xD756,12
C7×Q8Direct product of C7 and Q8562C7xQ856,10
C2×Dic7Direct product of C2 and Dic756C2xDic756,6

Groups of order 57

dρLabelID
C57Cyclic group571C5757,2
C19⋊C3The semidirect product of C19 and C3 acting faithfully193C19:C357,1

Groups of order 58

dρLabelID
C58Cyclic group581C5858,2
D29Dihedral group292+D2958,1

Groups of order 59

dρLabelID
C59Cyclic group591C5959,1

Groups of order 60

dρLabelID
C60Cyclic group601C6060,4
A5Alternating group on 5 letters; = SL2(𝔽4) = L2(4) = L2(5) = icosahedron/dodecahedron rotations; 1st non-abelian simple53+A560,5
D30Dihedral group; = C2×D15302+D3060,12
Dic15Dicyclic group; = C3Dic5602-Dic1560,3
C3⋊F5The semidirect product of C3 and F5 acting via F5/D5=C2154C3:F560,7
C2×C30Abelian group of type [2,30]60C2xC3060,13
S3×D5Direct product of S3 and D5154+S3xD560,8
C3×F5Direct product of C3 and F5154C3xF560,6
C5×A4Direct product of C5 and A4203C5xA460,9
C6×D5Direct product of C6 and D5302C6xD560,10
S3×C10Direct product of C10 and S3302S3xC1060,11
C5×Dic3Direct product of C5 and Dic3602C5xDic360,1
C3×Dic5Direct product of C3 and Dic5602C3xDic560,2
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