Aliases: Dic5.2S4, GL2(𝔽3)⋊1D5, SL2(𝔽3)⋊1D10, C2.7(D5×S4), C10.4(C2×S4), Q8⋊D15⋊4C2, Q8.4(S3×D5), (C5×Q8).4D6, C5⋊1(C4.3S4), Q8⋊2D5⋊3S3, Dic5.A4⋊3C2, (C5×GL2(𝔽3))⋊1C2, (C5×SL2(𝔽3))⋊1C22, SmallGroup(480,970)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C5×SL2(𝔽3) — GL2(𝔽3)⋊D5 |
| C5×SL2(𝔽3) — GL2(𝔽3)⋊D5 |
Generators and relations for GL2(𝔽3)⋊D5
G = < a,b,c,d,e,f | a4=c3=d2=e5=f2=1, b2=a2, bab-1=faf=dbd=a-1, cac-1=ab, dad=fbf=a2b, ae=ea, cbc-1=a, be=eb, dcd=c-1, ce=ec, fcf=ac, de=ed, df=fd, fef=e-1 >
Subgroups: 818 in 84 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C5, S3, C6, C8, C2×C4, D4, Q8, C23, D5, C10, C10, C12, D6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, SL2(𝔽3), D12, C5×S3, D15, C30, C8⋊C22, C5⋊2C8, C40, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, GL2(𝔽3), GL2(𝔽3), C4.A4, C3×Dic5, S3×C10, D30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q8⋊2D5, C4.3S4, C5⋊D12, C5×SL2(𝔽3), D40⋊C2, C5×GL2(𝔽3), Q8⋊D15, Dic5.A4, GL2(𝔽3)⋊D5
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, C2×S4, S3×D5, C4.3S4, D5×S4, GL2(𝔽3)⋊D5
Character table of GL2(𝔽3)⋊D5
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 20A | 20B | 30A | 30B | 40A | 40B | 40C | 40D | |
| size | 1 | 1 | 12 | 30 | 60 | 8 | 6 | 10 | 2 | 2 | 8 | 12 | 60 | 2 | 2 | 24 | 24 | 40 | 40 | 16 | 16 | 12 | 12 | 16 | 16 | 12 | 12 | 12 | 12 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | -2 | 0 | -1 | 2 | -2 | 2 | 2 | -1 | 0 | 0 | 2 | 2 | 0 | 0 | 1 | 1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 0 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | -1-√5/2 | -1+√5/2 | 2 | -2 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | orthogonal lifted from D10 |
| ρ8 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | -1+√5/2 | -1-√5/2 | 2 | 2 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ9 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | -1+√5/2 | -1-√5/2 | 2 | -2 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | orthogonal lifted from D10 |
| ρ10 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | -1-√5/2 | -1+√5/2 | 2 | 2 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ11 | 3 | 3 | 1 | 1 | -1 | 0 | -1 | -3 | 3 | 3 | 0 | -1 | 1 | 3 | 3 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from C2×S4 |
| ρ12 | 3 | 3 | 1 | -1 | 1 | 0 | -1 | 3 | 3 | 3 | 0 | -1 | -1 | 3 | 3 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ13 | 3 | 3 | -1 | -1 | -1 | 0 | -1 | 3 | 3 | 3 | 0 | 1 | 1 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from S4 |
| ρ14 | 3 | 3 | -1 | 1 | 1 | 0 | -1 | -3 | 3 | 3 | 0 | 1 | -1 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×S4 |
| ρ15 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 4 | 4 | 2 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.3S4 |
| ρ16 | 4 | 4 | 0 | 0 | 0 | -2 | 4 | 0 | -1-√5 | -1+√5 | -2 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | -1-√5 | -1+√5 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D5 |
| ρ17 | 4 | 4 | 0 | 0 | 0 | -2 | 4 | 0 | -1+√5 | -1-√5 | -2 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | -1+√5 | -1-√5 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D5 |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 1 | 0 | 0 | 4 | 4 | -1 | 0 | 0 | -4 | -4 | 0 | 0 | √3 | -√3 | 1 | 1 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.3S4 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 1 | 0 | 0 | 4 | 4 | -1 | 0 | 0 | -4 | -4 | 0 | 0 | -√3 | √3 | 1 | 1 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.3S4 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | -1-√5 | -1+√5 | 2 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ83ζ53-ζ83ζ52+ζ8ζ53-ζ8ζ52 | -ζ83ζ53+ζ83ζ52-ζ8ζ53+ζ8ζ52 | ζ87ζ54-ζ87ζ5+ζ85ζ54-ζ85ζ5 | ζ83ζ54-ζ83ζ5+ζ8ζ54-ζ8ζ5 | orthogonal faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | -1-√5 | -1+√5 | 2 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -ζ83ζ53+ζ83ζ52-ζ8ζ53+ζ8ζ52 | ζ83ζ53-ζ83ζ52+ζ8ζ53-ζ8ζ52 | ζ83ζ54-ζ83ζ5+ζ8ζ54-ζ8ζ5 | ζ87ζ54-ζ87ζ5+ζ85ζ54-ζ85ζ5 | orthogonal faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | -1+√5 | -1-√5 | 2 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ83ζ54-ζ83ζ5+ζ8ζ54-ζ8ζ5 | ζ87ζ54-ζ87ζ5+ζ85ζ54-ζ85ζ5 | ζ83ζ53-ζ83ζ52+ζ8ζ53-ζ8ζ52 | -ζ83ζ53+ζ83ζ52-ζ8ζ53+ζ8ζ52 | orthogonal faithful |
| ρ23 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | -1+√5 | -1-√5 | 2 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ87ζ54-ζ87ζ5+ζ85ζ54-ζ85ζ5 | ζ83ζ54-ζ83ζ5+ζ8ζ54-ζ8ζ5 | -ζ83ζ53+ζ83ζ52-ζ8ζ53+ζ8ζ52 | ζ83ζ53-ζ83ζ52+ζ8ζ53-ζ8ζ52 | orthogonal faithful |
| ρ24 | 6 | 6 | 2 | 0 | 0 | 0 | -2 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | -2 | 0 | -3+3√5/2 | -3-3√5/2 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | orthogonal lifted from D5×S4 |
| ρ25 | 6 | 6 | -2 | 0 | 0 | 0 | -2 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 2 | 0 | -3-3√5/2 | -3+3√5/2 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D5×S4 |
| ρ26 | 6 | 6 | 2 | 0 | 0 | 0 | -2 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | -2 | 0 | -3-3√5/2 | -3+3√5/2 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | orthogonal lifted from D5×S4 |
| ρ27 | 6 | 6 | -2 | 0 | 0 | 0 | -2 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 2 | 0 | -3+3√5/2 | -3-3√5/2 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D5×S4 |
| ρ28 | 8 | -8 | 0 | 0 | 0 | 2 | 0 | 0 | -2+2√5 | -2-2√5 | -2 | 0 | 0 | 2+2√5 | 2-2√5 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
| ρ29 | 8 | -8 | 0 | 0 | 0 | 2 | 0 | 0 | -2-2√5 | -2+2√5 | -2 | 0 | 0 | 2-2√5 | 2+2√5 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
►On 80 points
Generators in S80
(1 54 12 69)(2 55 13 70)(3 51 14 66)(4 52 15 67)(5 53 11 68)(6 19 60 64)(7 20 56 65)(8 16 57 61)(9 17 58 62)(10 18 59 63)(21 34 78 47)(22 35 79 48)(23 31 80 49)(24 32 76 50)(25 33 77 46)(26 45 74 40)(27 41 75 36)(28 42 71 37)(29 43 72 38)(30 44 73 39)
(1 42 12 37)(2 43 13 38)(3 44 14 39)(4 45 15 40)(5 41 11 36)(6 21 60 78)(7 22 56 79)(8 23 57 80)(9 24 58 76)(10 25 59 77)(16 49 61 31)(17 50 62 32)(18 46 63 33)(19 47 64 34)(20 48 65 35)(26 67 74 52)(27 68 75 53)(28 69 71 54)(29 70 72 55)(30 66 73 51)
(16 23 49)(17 24 50)(18 25 46)(19 21 47)(20 22 48)(26 52 45)(27 53 41)(28 54 42)(29 55 43)(30 51 44)(31 61 80)(32 62 76)(33 63 77)(34 64 78)(35 65 79)(36 75 68)(37 71 69)(38 72 70)(39 73 66)(40 74 67)
(6 60)(7 56)(8 57)(9 58)(10 59)(16 23)(17 24)(18 25)(19 21)(20 22)(26 74)(27 75)(28 71)(29 72)(30 73)(36 53)(37 54)(38 55)(39 51)(40 52)(41 68)(42 69)(43 70)(44 66)(45 67)(61 80)(62 76)(63 77)(64 78)(65 79)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)
(1 46)(2 50)(3 49)(4 48)(5 47)(6 75)(7 74)(8 73)(9 72)(10 71)(11 34)(12 33)(13 32)(14 31)(15 35)(16 44)(17 43)(18 42)(19 41)(20 45)(21 68)(22 67)(23 66)(24 70)(25 69)(26 56)(27 60)(28 59)(29 58)(30 57)(36 64)(37 63)(38 62)(39 61)(40 65)(51 80)(52 79)(53 78)(54 77)(55 76)
G:=sub<Sym(80)| (1,54,12,69)(2,55,13,70)(3,51,14,66)(4,52,15,67)(5,53,11,68)(6,19,60,64)(7,20,56,65)(8,16,57,61)(9,17,58,62)(10,18,59,63)(21,34,78,47)(22,35,79,48)(23,31,80,49)(24,32,76,50)(25,33,77,46)(26,45,74,40)(27,41,75,36)(28,42,71,37)(29,43,72,38)(30,44,73,39), (1,42,12,37)(2,43,13,38)(3,44,14,39)(4,45,15,40)(5,41,11,36)(6,21,60,78)(7,22,56,79)(8,23,57,80)(9,24,58,76)(10,25,59,77)(16,49,61,31)(17,50,62,32)(18,46,63,33)(19,47,64,34)(20,48,65,35)(26,67,74,52)(27,68,75,53)(28,69,71,54)(29,70,72,55)(30,66,73,51), (16,23,49)(17,24,50)(18,25,46)(19,21,47)(20,22,48)(26,52,45)(27,53,41)(28,54,42)(29,55,43)(30,51,44)(31,61,80)(32,62,76)(33,63,77)(34,64,78)(35,65,79)(36,75,68)(37,71,69)(38,72,70)(39,73,66)(40,74,67), (6,60)(7,56)(8,57)(9,58)(10,59)(16,23)(17,24)(18,25)(19,21)(20,22)(26,74)(27,75)(28,71)(29,72)(30,73)(36,53)(37,54)(38,55)(39,51)(40,52)(41,68)(42,69)(43,70)(44,66)(45,67)(61,80)(62,76)(63,77)(64,78)(65,79), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,46)(2,50)(3,49)(4,48)(5,47)(6,75)(7,74)(8,73)(9,72)(10,71)(11,34)(12,33)(13,32)(14,31)(15,35)(16,44)(17,43)(18,42)(19,41)(20,45)(21,68)(22,67)(23,66)(24,70)(25,69)(26,56)(27,60)(28,59)(29,58)(30,57)(36,64)(37,63)(38,62)(39,61)(40,65)(51,80)(52,79)(53,78)(54,77)(55,76)>;
G:=Group( (1,54,12,69)(2,55,13,70)(3,51,14,66)(4,52,15,67)(5,53,11,68)(6,19,60,64)(7,20,56,65)(8,16,57,61)(9,17,58,62)(10,18,59,63)(21,34,78,47)(22,35,79,48)(23,31,80,49)(24,32,76,50)(25,33,77,46)(26,45,74,40)(27,41,75,36)(28,42,71,37)(29,43,72,38)(30,44,73,39), (1,42,12,37)(2,43,13,38)(3,44,14,39)(4,45,15,40)(5,41,11,36)(6,21,60,78)(7,22,56,79)(8,23,57,80)(9,24,58,76)(10,25,59,77)(16,49,61,31)(17,50,62,32)(18,46,63,33)(19,47,64,34)(20,48,65,35)(26,67,74,52)(27,68,75,53)(28,69,71,54)(29,70,72,55)(30,66,73,51), (16,23,49)(17,24,50)(18,25,46)(19,21,47)(20,22,48)(26,52,45)(27,53,41)(28,54,42)(29,55,43)(30,51,44)(31,61,80)(32,62,76)(33,63,77)(34,64,78)(35,65,79)(36,75,68)(37,71,69)(38,72,70)(39,73,66)(40,74,67), (6,60)(7,56)(8,57)(9,58)(10,59)(16,23)(17,24)(18,25)(19,21)(20,22)(26,74)(27,75)(28,71)(29,72)(30,73)(36,53)(37,54)(38,55)(39,51)(40,52)(41,68)(42,69)(43,70)(44,66)(45,67)(61,80)(62,76)(63,77)(64,78)(65,79), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,46)(2,50)(3,49)(4,48)(5,47)(6,75)(7,74)(8,73)(9,72)(10,71)(11,34)(12,33)(13,32)(14,31)(15,35)(16,44)(17,43)(18,42)(19,41)(20,45)(21,68)(22,67)(23,66)(24,70)(25,69)(26,56)(27,60)(28,59)(29,58)(30,57)(36,64)(37,63)(38,62)(39,61)(40,65)(51,80)(52,79)(53,78)(54,77)(55,76) );
G=PermutationGroup([[(1,54,12,69),(2,55,13,70),(3,51,14,66),(4,52,15,67),(5,53,11,68),(6,19,60,64),(7,20,56,65),(8,16,57,61),(9,17,58,62),(10,18,59,63),(21,34,78,47),(22,35,79,48),(23,31,80,49),(24,32,76,50),(25,33,77,46),(26,45,74,40),(27,41,75,36),(28,42,71,37),(29,43,72,38),(30,44,73,39)], [(1,42,12,37),(2,43,13,38),(3,44,14,39),(4,45,15,40),(5,41,11,36),(6,21,60,78),(7,22,56,79),(8,23,57,80),(9,24,58,76),(10,25,59,77),(16,49,61,31),(17,50,62,32),(18,46,63,33),(19,47,64,34),(20,48,65,35),(26,67,74,52),(27,68,75,53),(28,69,71,54),(29,70,72,55),(30,66,73,51)], [(16,23,49),(17,24,50),(18,25,46),(19,21,47),(20,22,48),(26,52,45),(27,53,41),(28,54,42),(29,55,43),(30,51,44),(31,61,80),(32,62,76),(33,63,77),(34,64,78),(35,65,79),(36,75,68),(37,71,69),(38,72,70),(39,73,66),(40,74,67)], [(6,60),(7,56),(8,57),(9,58),(10,59),(16,23),(17,24),(18,25),(19,21),(20,22),(26,74),(27,75),(28,71),(29,72),(30,73),(36,53),(37,54),(38,55),(39,51),(40,52),(41,68),(42,69),(43,70),(44,66),(45,67),(61,80),(62,76),(63,77),(64,78),(65,79)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80)], [(1,46),(2,50),(3,49),(4,48),(5,47),(6,75),(7,74),(8,73),(9,72),(10,71),(11,34),(12,33),(13,32),(14,31),(15,35),(16,44),(17,43),(18,42),(19,41),(20,45),(21,68),(22,67),(23,66),(24,70),(25,69),(26,56),(27,60),(28,59),(29,58),(30,57),(36,64),(37,63),(38,62),(39,61),(40,65),(51,80),(52,79),(53,78),(54,77),(55,76)]])
Matrix representation of GL2(𝔽3)⋊D5 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 240 | 189 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 173 | 0 | 0 | 0 | 0 |
| 64 | 225 | 0 | 0 | 0 | 0 |
| 0 | 0 | 99 | 99 | 99 | 0 |
| 0 | 0 | 99 | 142 | 0 | 99 |
| 0 | 0 | 99 | 0 | 142 | 142 |
| 0 | 0 | 0 | 99 | 142 | 99 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,240,0,0,0,0,0,0,240,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,240],[0,240,0,0,0,0,1,189,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,64,0,0,0,0,173,225,0,0,0,0,0,0,99,99,99,0,0,0,99,142,0,99,0,0,99,0,142,142,0,0,0,99,142,99] >;
GL2(𝔽3)⋊D5 in GAP, Magma, Sage, TeX
{\rm GL}_2({\mathbb F}_3)\rtimes D_5 % in TeX
G:=Group("GL(2,3):D5"); // GroupNames label
G:=SmallGroup(480,970);
// by ID
G=gap.SmallGroup(480,970);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,1680,3389,93,1347,2111,3168,172,1272,1909,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=c^3=d^2=e^5=f^2=1,b^2=a^2,b*a*b^-1=f*a*f=d*b*d=a^-1,c*a*c^-1=a*b,d*a*d=f*b*f=a^2*b,a*e=e*a,c*b*c^-1=a,b*e=e*b,d*c*d=c^-1,c*e=e*c,f*c*f=a*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations
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