Aliases: Dic5.1S4, CSU2(𝔽3)⋊1D5, SL2(𝔽3).1D10, C2.4(D5×S4), C10.1(C2×S4), Q8.1(S3×D5), (C5×Q8).1D6, C5⋊1(C4.S4), Q8.D15⋊3C2, Q8⋊2D5.1S3, Dic5.A4.1C2, (C5×CSU2(𝔽3))⋊3C2, (C5×SL2(𝔽3)).1C22, SmallGroup(480,967)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C5×SL2(𝔽3) — CSU2(𝔽3)⋊D5 |
| C5×SL2(𝔽3) — CSU2(𝔽3)⋊D5 |
Generators and relations for CSU2(𝔽3)⋊D5
G = < a,b,c,d,e,f | a4=c3=e5=f2=1, b2=d2=a2, bab-1=faf=dbd-1=a-1, cac-1=ab, dad-1=fbf=a2b, ae=ea, cbc-1=a, be=eb, dcd-1=c-1, ce=ec, fcf=ac, de=ed, df=fd, fef=e-1 >
Subgroups: 514 in 72 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C8, C2×C4, D4, Q8, Q8, D5, C10, Dic3, C12, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, Dic5, C20, D10, SL2(𝔽3), Dic6, C30, C8.C22, C5⋊2C8, C40, Dic10, C4×D5, D20, C5×Q8, C5×Q8, CSU2(𝔽3), CSU2(𝔽3), C4.A4, C5×Dic3, C3×Dic5, Dic15, C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q8⋊2D5, C4.S4, C15⋊Q8, C5×SL2(𝔽3), Q16⋊D5, C5×CSU2(𝔽3), Q8.D15, Dic5.A4, CSU2(𝔽3)⋊D5
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, C2×S4, S3×D5, C4.S4, D5×S4, CSU2(𝔽3)⋊D5
Character table of CSU2(𝔽3)⋊D5
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | 40A | 40B | 40C | 40D | |
| size | 1 | 1 | 30 | 8 | 6 | 10 | 12 | 60 | 2 | 2 | 8 | 12 | 60 | 2 | 2 | 40 | 40 | 16 | 16 | 12 | 12 | 24 | 24 | 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 | -2 | -1 | 2 | -2 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 2 | 2 | 1 | 1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 0 | 2 | 2 | 0 | 2 | 0 | -1-√5/2 | -1+√5/2 | 2 | 2 | 0 | -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 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ8 | 2 | 2 | 0 | 2 | 2 | 0 | -2 | 0 | -1+√5/2 | -1-√5/2 | 2 | -2 | 0 | -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 | 1-√5/2 | 1-√5/2 | orthogonal lifted from D10 |
| ρ9 | 2 | 2 | 0 | 2 | 2 | 0 | 2 | 0 | -1+√5/2 | -1-√5/2 | 2 | 2 | 0 | -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 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ10 | 2 | 2 | 0 | 2 | 2 | 0 | -2 | 0 | -1-√5/2 | -1+√5/2 | 2 | -2 | 0 | -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 | 1+√5/2 | 1+√5/2 | orthogonal lifted from D10 |
| ρ11 | 3 | 3 | 1 | 0 | -1 | -3 | 1 | -1 | 3 | 3 | 0 | -1 | 1 | 3 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from C2×S4 |
| ρ12 | 3 | 3 | 1 | 0 | -1 | -3 | -1 | 1 | 3 | 3 | 0 | 1 | -1 | 3 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×S4 |
| ρ13 | 3 | 3 | -1 | 0 | -1 | 3 | -1 | -1 | 3 | 3 | 0 | 1 | 1 | 3 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from S4 |
| ρ14 | 3 | 3 | -1 | 0 | -1 | 3 | 1 | 1 | 3 | 3 | 0 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ15 | 4 | 4 | 0 | -2 | 4 | 0 | 0 | 0 | -1-√5 | -1+√5 | -2 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 1-√5/2 | 1+√5/2 | -1+√5 | -1-√5 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D5 |
| ρ16 | 4 | 4 | 0 | -2 | 4 | 0 | 0 | 0 | -1+√5 | -1-√5 | -2 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | 1+√5/2 | 1-√5/2 | -1-√5 | -1+√5 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D5 |
| ρ17 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 4 | 4 | 2 | 0 | 0 | -4 | -4 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from C4.S4, Schur index 2 |
| ρ18 | 4 | -4 | 0 | 1 | 0 | 0 | 0 | 0 | 4 | 4 | -1 | 0 | 0 | -4 | -4 | √3 | -√3 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | symplectic lifted from C4.S4, Schur index 2 |
| ρ19 | 4 | -4 | 0 | 1 | 0 | 0 | 0 | 0 | 4 | 4 | -1 | 0 | 0 | -4 | -4 | -√3 | √3 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | symplectic lifted from C4.S4, Schur index 2 |
| ρ20 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 2 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -ζ83ζ53+ζ83ζ52+ζ8ζ53-ζ8ζ52 | ζ83ζ53-ζ83ζ52-ζ8ζ53+ζ8ζ52 | ζ83ζ54-ζ83ζ5-ζ8ζ54+ζ8ζ5 | ζ87ζ54-ζ87ζ5-ζ85ζ54+ζ85ζ5 | complex faithful |
| ρ21 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 2 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ87ζ54-ζ87ζ5-ζ85ζ54+ζ85ζ5 | ζ83ζ54-ζ83ζ5-ζ8ζ54+ζ8ζ5 | -ζ83ζ53+ζ83ζ52+ζ8ζ53-ζ8ζ52 | ζ83ζ53-ζ83ζ52-ζ8ζ53+ζ8ζ52 | complex faithful |
| ρ22 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 2 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ83ζ53-ζ83ζ52-ζ8ζ53+ζ8ζ52 | -ζ83ζ53+ζ83ζ52+ζ8ζ53-ζ8ζ52 | ζ87ζ54-ζ87ζ5-ζ85ζ54+ζ85ζ5 | ζ83ζ54-ζ83ζ5-ζ8ζ54+ζ8ζ5 | complex faithful |
| ρ23 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 2 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ83ζ54-ζ83ζ5-ζ8ζ54+ζ8ζ5 | ζ87ζ54-ζ87ζ5-ζ85ζ54+ζ85ζ5 | ζ83ζ53-ζ83ζ52-ζ8ζ53+ζ8ζ52 | -ζ83ζ53+ζ83ζ52+ζ8ζ53-ζ8ζ52 | complex faithful |
| ρ24 | 6 | 6 | 0 | 0 | -2 | 0 | 2 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | -2 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | 0 | 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 | orthogonal lifted from D5×S4 |
| ρ25 | 6 | 6 | 0 | 0 | -2 | 0 | -2 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 2 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | 0 | 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 | orthogonal lifted from D5×S4 |
| ρ26 | 6 | 6 | 0 | 0 | -2 | 0 | -2 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 2 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | 0 | 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 | orthogonal lifted from D5×S4 |
| ρ27 | 6 | 6 | 0 | 0 | -2 | 0 | 2 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | -2 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | 0 | 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 | orthogonal lifted from D5×S4 |
| ρ28 | 8 | -8 | 0 | 2 | 0 | 0 | 0 | 0 | -2-2√5 | -2+2√5 | -2 | 0 | 0 | 2-2√5 | 2+2√5 | 0 | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ29 | 8 | -8 | 0 | 2 | 0 | 0 | 0 | 0 | -2+2√5 | -2-2√5 | -2 | 0 | 0 | 2+2√5 | 2-2√5 | 0 | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 160 points
Generators in S160
(1 54 12 42)(2 55 13 43)(3 51 14 44)(4 52 15 45)(5 53 11 41)(6 158 140 21)(7 159 136 22)(8 160 137 23)(9 156 138 24)(10 157 139 25)(16 31 141 129)(17 32 142 130)(18 33 143 126)(19 34 144 127)(20 35 145 128)(26 69 50 40)(27 70 46 36)(28 66 47 37)(29 67 48 38)(30 68 49 39)(56 73 94 111)(57 74 95 112)(58 75 91 113)(59 71 92 114)(60 72 93 115)(61 83 106 78)(62 84 107 79)(63 85 108 80)(64 81 109 76)(65 82 110 77)(86 123 152 117)(87 124 153 118)(88 125 154 119)(89 121 155 120)(90 122 151 116)(96 105 150 134)(97 101 146 135)(98 102 147 131)(99 103 148 132)(100 104 149 133)
(1 66 12 37)(2 67 13 38)(3 68 14 39)(4 69 15 40)(5 70 11 36)(6 144 140 19)(7 145 136 20)(8 141 137 16)(9 142 138 17)(10 143 139 18)(21 127 158 34)(22 128 159 35)(23 129 160 31)(24 130 156 32)(25 126 157 33)(26 45 50 52)(27 41 46 53)(28 42 47 54)(29 43 48 55)(30 44 49 51)(56 65 94 110)(57 61 95 106)(58 62 91 107)(59 63 92 108)(60 64 93 109)(71 80 114 85)(72 76 115 81)(73 77 111 82)(74 78 112 83)(75 79 113 84)(86 103 152 132)(87 104 153 133)(88 105 154 134)(89 101 155 135)(90 102 151 131)(96 119 150 125)(97 120 146 121)(98 116 147 122)(99 117 148 123)(100 118 149 124)
(16 129 23)(17 130 24)(18 126 25)(19 127 21)(20 128 22)(26 52 69)(27 53 70)(28 54 66)(29 55 67)(30 51 68)(31 160 141)(32 156 142)(33 157 143)(34 158 144)(35 159 145)(36 46 41)(37 47 42)(38 48 43)(39 49 44)(40 50 45)(56 111 77)(57 112 78)(58 113 79)(59 114 80)(60 115 76)(71 85 92)(72 81 93)(73 82 94)(74 83 95)(75 84 91)(86 132 123)(87 133 124)(88 134 125)(89 135 121)(90 131 122)(101 120 155)(102 116 151)(103 117 152)(104 118 153)(105 119 154)
(1 65 12 110)(2 61 13 106)(3 62 14 107)(4 63 15 108)(5 64 11 109)(6 100 140 149)(7 96 136 150)(8 97 137 146)(9 98 138 147)(10 99 139 148)(16 101 141 135)(17 102 142 131)(18 103 143 132)(19 104 144 133)(20 105 145 134)(21 118 158 124)(22 119 159 125)(23 120 160 121)(24 116 156 122)(25 117 157 123)(26 114 50 71)(27 115 46 72)(28 111 47 73)(29 112 48 74)(30 113 49 75)(31 89 129 155)(32 90 130 151)(33 86 126 152)(34 87 127 153)(35 88 128 154)(36 81 70 76)(37 82 66 77)(38 83 67 78)(39 84 68 79)(40 85 69 80)(41 93 53 60)(42 94 54 56)(43 95 55 57)(44 91 51 58)(45 92 52 59)
(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)(81 82 83 84 85)(86 87 88 89 90)(91 92 93 94 95)(96 97 98 99 100)(101 102 103 104 105)(106 107 108 109 110)(111 112 113 114 115)(116 117 118 119 120)(121 122 123 124 125)(126 127 128 129 130)(131 132 133 134 135)(136 137 138 139 140)(141 142 143 144 145)(146 147 148 149 150)(151 152 153 154 155)(156 157 158 159 160)
(1 86)(2 90)(3 89)(4 88)(5 87)(6 72)(7 71)(8 75)(9 74)(10 73)(11 153)(12 152)(13 151)(14 155)(15 154)(16 79)(17 78)(18 77)(19 76)(20 80)(21 93)(22 92)(23 91)(24 95)(25 94)(26 96)(27 100)(28 99)(29 98)(30 97)(31 107)(32 106)(33 110)(34 109)(35 108)(36 104)(37 103)(38 102)(39 101)(40 105)(41 124)(42 123)(43 122)(44 121)(45 125)(46 149)(47 148)(48 147)(49 146)(50 150)(51 120)(52 119)(53 118)(54 117)(55 116)(56 157)(57 156)(58 160)(59 159)(60 158)(61 130)(62 129)(63 128)(64 127)(65 126)(66 132)(67 131)(68 135)(69 134)(70 133)(81 144)(82 143)(83 142)(84 141)(85 145)(111 139)(112 138)(113 137)(114 136)(115 140)
G:=sub<Sym(160)| (1,54,12,42)(2,55,13,43)(3,51,14,44)(4,52,15,45)(5,53,11,41)(6,158,140,21)(7,159,136,22)(8,160,137,23)(9,156,138,24)(10,157,139,25)(16,31,141,129)(17,32,142,130)(18,33,143,126)(19,34,144,127)(20,35,145,128)(26,69,50,40)(27,70,46,36)(28,66,47,37)(29,67,48,38)(30,68,49,39)(56,73,94,111)(57,74,95,112)(58,75,91,113)(59,71,92,114)(60,72,93,115)(61,83,106,78)(62,84,107,79)(63,85,108,80)(64,81,109,76)(65,82,110,77)(86,123,152,117)(87,124,153,118)(88,125,154,119)(89,121,155,120)(90,122,151,116)(96,105,150,134)(97,101,146,135)(98,102,147,131)(99,103,148,132)(100,104,149,133), (1,66,12,37)(2,67,13,38)(3,68,14,39)(4,69,15,40)(5,70,11,36)(6,144,140,19)(7,145,136,20)(8,141,137,16)(9,142,138,17)(10,143,139,18)(21,127,158,34)(22,128,159,35)(23,129,160,31)(24,130,156,32)(25,126,157,33)(26,45,50,52)(27,41,46,53)(28,42,47,54)(29,43,48,55)(30,44,49,51)(56,65,94,110)(57,61,95,106)(58,62,91,107)(59,63,92,108)(60,64,93,109)(71,80,114,85)(72,76,115,81)(73,77,111,82)(74,78,112,83)(75,79,113,84)(86,103,152,132)(87,104,153,133)(88,105,154,134)(89,101,155,135)(90,102,151,131)(96,119,150,125)(97,120,146,121)(98,116,147,122)(99,117,148,123)(100,118,149,124), (16,129,23)(17,130,24)(18,126,25)(19,127,21)(20,128,22)(26,52,69)(27,53,70)(28,54,66)(29,55,67)(30,51,68)(31,160,141)(32,156,142)(33,157,143)(34,158,144)(35,159,145)(36,46,41)(37,47,42)(38,48,43)(39,49,44)(40,50,45)(56,111,77)(57,112,78)(58,113,79)(59,114,80)(60,115,76)(71,85,92)(72,81,93)(73,82,94)(74,83,95)(75,84,91)(86,132,123)(87,133,124)(88,134,125)(89,135,121)(90,131,122)(101,120,155)(102,116,151)(103,117,152)(104,118,153)(105,119,154), (1,65,12,110)(2,61,13,106)(3,62,14,107)(4,63,15,108)(5,64,11,109)(6,100,140,149)(7,96,136,150)(8,97,137,146)(9,98,138,147)(10,99,139,148)(16,101,141,135)(17,102,142,131)(18,103,143,132)(19,104,144,133)(20,105,145,134)(21,118,158,124)(22,119,159,125)(23,120,160,121)(24,116,156,122)(25,117,157,123)(26,114,50,71)(27,115,46,72)(28,111,47,73)(29,112,48,74)(30,113,49,75)(31,89,129,155)(32,90,130,151)(33,86,126,152)(34,87,127,153)(35,88,128,154)(36,81,70,76)(37,82,66,77)(38,83,67,78)(39,84,68,79)(40,85,69,80)(41,93,53,60)(42,94,54,56)(43,95,55,57)(44,91,51,58)(45,92,52,59), (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)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160), (1,86)(2,90)(3,89)(4,88)(5,87)(6,72)(7,71)(8,75)(9,74)(10,73)(11,153)(12,152)(13,151)(14,155)(15,154)(16,79)(17,78)(18,77)(19,76)(20,80)(21,93)(22,92)(23,91)(24,95)(25,94)(26,96)(27,100)(28,99)(29,98)(30,97)(31,107)(32,106)(33,110)(34,109)(35,108)(36,104)(37,103)(38,102)(39,101)(40,105)(41,124)(42,123)(43,122)(44,121)(45,125)(46,149)(47,148)(48,147)(49,146)(50,150)(51,120)(52,119)(53,118)(54,117)(55,116)(56,157)(57,156)(58,160)(59,159)(60,158)(61,130)(62,129)(63,128)(64,127)(65,126)(66,132)(67,131)(68,135)(69,134)(70,133)(81,144)(82,143)(83,142)(84,141)(85,145)(111,139)(112,138)(113,137)(114,136)(115,140)>;
G:=Group( (1,54,12,42)(2,55,13,43)(3,51,14,44)(4,52,15,45)(5,53,11,41)(6,158,140,21)(7,159,136,22)(8,160,137,23)(9,156,138,24)(10,157,139,25)(16,31,141,129)(17,32,142,130)(18,33,143,126)(19,34,144,127)(20,35,145,128)(26,69,50,40)(27,70,46,36)(28,66,47,37)(29,67,48,38)(30,68,49,39)(56,73,94,111)(57,74,95,112)(58,75,91,113)(59,71,92,114)(60,72,93,115)(61,83,106,78)(62,84,107,79)(63,85,108,80)(64,81,109,76)(65,82,110,77)(86,123,152,117)(87,124,153,118)(88,125,154,119)(89,121,155,120)(90,122,151,116)(96,105,150,134)(97,101,146,135)(98,102,147,131)(99,103,148,132)(100,104,149,133), (1,66,12,37)(2,67,13,38)(3,68,14,39)(4,69,15,40)(5,70,11,36)(6,144,140,19)(7,145,136,20)(8,141,137,16)(9,142,138,17)(10,143,139,18)(21,127,158,34)(22,128,159,35)(23,129,160,31)(24,130,156,32)(25,126,157,33)(26,45,50,52)(27,41,46,53)(28,42,47,54)(29,43,48,55)(30,44,49,51)(56,65,94,110)(57,61,95,106)(58,62,91,107)(59,63,92,108)(60,64,93,109)(71,80,114,85)(72,76,115,81)(73,77,111,82)(74,78,112,83)(75,79,113,84)(86,103,152,132)(87,104,153,133)(88,105,154,134)(89,101,155,135)(90,102,151,131)(96,119,150,125)(97,120,146,121)(98,116,147,122)(99,117,148,123)(100,118,149,124), (16,129,23)(17,130,24)(18,126,25)(19,127,21)(20,128,22)(26,52,69)(27,53,70)(28,54,66)(29,55,67)(30,51,68)(31,160,141)(32,156,142)(33,157,143)(34,158,144)(35,159,145)(36,46,41)(37,47,42)(38,48,43)(39,49,44)(40,50,45)(56,111,77)(57,112,78)(58,113,79)(59,114,80)(60,115,76)(71,85,92)(72,81,93)(73,82,94)(74,83,95)(75,84,91)(86,132,123)(87,133,124)(88,134,125)(89,135,121)(90,131,122)(101,120,155)(102,116,151)(103,117,152)(104,118,153)(105,119,154), (1,65,12,110)(2,61,13,106)(3,62,14,107)(4,63,15,108)(5,64,11,109)(6,100,140,149)(7,96,136,150)(8,97,137,146)(9,98,138,147)(10,99,139,148)(16,101,141,135)(17,102,142,131)(18,103,143,132)(19,104,144,133)(20,105,145,134)(21,118,158,124)(22,119,159,125)(23,120,160,121)(24,116,156,122)(25,117,157,123)(26,114,50,71)(27,115,46,72)(28,111,47,73)(29,112,48,74)(30,113,49,75)(31,89,129,155)(32,90,130,151)(33,86,126,152)(34,87,127,153)(35,88,128,154)(36,81,70,76)(37,82,66,77)(38,83,67,78)(39,84,68,79)(40,85,69,80)(41,93,53,60)(42,94,54,56)(43,95,55,57)(44,91,51,58)(45,92,52,59), (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)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160), (1,86)(2,90)(3,89)(4,88)(5,87)(6,72)(7,71)(8,75)(9,74)(10,73)(11,153)(12,152)(13,151)(14,155)(15,154)(16,79)(17,78)(18,77)(19,76)(20,80)(21,93)(22,92)(23,91)(24,95)(25,94)(26,96)(27,100)(28,99)(29,98)(30,97)(31,107)(32,106)(33,110)(34,109)(35,108)(36,104)(37,103)(38,102)(39,101)(40,105)(41,124)(42,123)(43,122)(44,121)(45,125)(46,149)(47,148)(48,147)(49,146)(50,150)(51,120)(52,119)(53,118)(54,117)(55,116)(56,157)(57,156)(58,160)(59,159)(60,158)(61,130)(62,129)(63,128)(64,127)(65,126)(66,132)(67,131)(68,135)(69,134)(70,133)(81,144)(82,143)(83,142)(84,141)(85,145)(111,139)(112,138)(113,137)(114,136)(115,140) );
G=PermutationGroup([[(1,54,12,42),(2,55,13,43),(3,51,14,44),(4,52,15,45),(5,53,11,41),(6,158,140,21),(7,159,136,22),(8,160,137,23),(9,156,138,24),(10,157,139,25),(16,31,141,129),(17,32,142,130),(18,33,143,126),(19,34,144,127),(20,35,145,128),(26,69,50,40),(27,70,46,36),(28,66,47,37),(29,67,48,38),(30,68,49,39),(56,73,94,111),(57,74,95,112),(58,75,91,113),(59,71,92,114),(60,72,93,115),(61,83,106,78),(62,84,107,79),(63,85,108,80),(64,81,109,76),(65,82,110,77),(86,123,152,117),(87,124,153,118),(88,125,154,119),(89,121,155,120),(90,122,151,116),(96,105,150,134),(97,101,146,135),(98,102,147,131),(99,103,148,132),(100,104,149,133)], [(1,66,12,37),(2,67,13,38),(3,68,14,39),(4,69,15,40),(5,70,11,36),(6,144,140,19),(7,145,136,20),(8,141,137,16),(9,142,138,17),(10,143,139,18),(21,127,158,34),(22,128,159,35),(23,129,160,31),(24,130,156,32),(25,126,157,33),(26,45,50,52),(27,41,46,53),(28,42,47,54),(29,43,48,55),(30,44,49,51),(56,65,94,110),(57,61,95,106),(58,62,91,107),(59,63,92,108),(60,64,93,109),(71,80,114,85),(72,76,115,81),(73,77,111,82),(74,78,112,83),(75,79,113,84),(86,103,152,132),(87,104,153,133),(88,105,154,134),(89,101,155,135),(90,102,151,131),(96,119,150,125),(97,120,146,121),(98,116,147,122),(99,117,148,123),(100,118,149,124)], [(16,129,23),(17,130,24),(18,126,25),(19,127,21),(20,128,22),(26,52,69),(27,53,70),(28,54,66),(29,55,67),(30,51,68),(31,160,141),(32,156,142),(33,157,143),(34,158,144),(35,159,145),(36,46,41),(37,47,42),(38,48,43),(39,49,44),(40,50,45),(56,111,77),(57,112,78),(58,113,79),(59,114,80),(60,115,76),(71,85,92),(72,81,93),(73,82,94),(74,83,95),(75,84,91),(86,132,123),(87,133,124),(88,134,125),(89,135,121),(90,131,122),(101,120,155),(102,116,151),(103,117,152),(104,118,153),(105,119,154)], [(1,65,12,110),(2,61,13,106),(3,62,14,107),(4,63,15,108),(5,64,11,109),(6,100,140,149),(7,96,136,150),(8,97,137,146),(9,98,138,147),(10,99,139,148),(16,101,141,135),(17,102,142,131),(18,103,143,132),(19,104,144,133),(20,105,145,134),(21,118,158,124),(22,119,159,125),(23,120,160,121),(24,116,156,122),(25,117,157,123),(26,114,50,71),(27,115,46,72),(28,111,47,73),(29,112,48,74),(30,113,49,75),(31,89,129,155),(32,90,130,151),(33,86,126,152),(34,87,127,153),(35,88,128,154),(36,81,70,76),(37,82,66,77),(38,83,67,78),(39,84,68,79),(40,85,69,80),(41,93,53,60),(42,94,54,56),(43,95,55,57),(44,91,51,58),(45,92,52,59)], [(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),(81,82,83,84,85),(86,87,88,89,90),(91,92,93,94,95),(96,97,98,99,100),(101,102,103,104,105),(106,107,108,109,110),(111,112,113,114,115),(116,117,118,119,120),(121,122,123,124,125),(126,127,128,129,130),(131,132,133,134,135),(136,137,138,139,140),(141,142,143,144,145),(146,147,148,149,150),(151,152,153,154,155),(156,157,158,159,160)], [(1,86),(2,90),(3,89),(4,88),(5,87),(6,72),(7,71),(8,75),(9,74),(10,73),(11,153),(12,152),(13,151),(14,155),(15,154),(16,79),(17,78),(18,77),(19,76),(20,80),(21,93),(22,92),(23,91),(24,95),(25,94),(26,96),(27,100),(28,99),(29,98),(30,97),(31,107),(32,106),(33,110),(34,109),(35,108),(36,104),(37,103),(38,102),(39,101),(40,105),(41,124),(42,123),(43,122),(44,121),(45,125),(46,149),(47,148),(48,147),(49,146),(50,150),(51,120),(52,119),(53,118),(54,117),(55,116),(56,157),(57,156),(58,160),(59,159),(60,158),(61,130),(62,129),(63,128),(64,127),(65,126),(66,132),(67,131),(68,135),(69,134),(70,133),(81,144),(82,143),(83,142),(84,141),(85,145),(111,139),(112,138),(113,137),(114,136),(115,140)]])
Matrix representation of CSU2(𝔽3)⋊D5 ►in GL8(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 | 240 | 239 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 | 240 | 239 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
| 240 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 2 |
| 0 | 0 | 0 | 0 | 240 | 240 | 0 | 240 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 228 | 0 | 62 | 62 |
| 0 | 0 | 0 | 0 | 0 | 62 | 228 | 62 |
| 0 | 0 | 0 | 0 | 62 | 228 | 0 | 62 |
| 0 | 0 | 0 | 0 | 13 | 13 | 13 | 192 |
| 240 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 1 | 0 | 0 | 0 | 0 |
| 50 | 0 | 190 | 0 | 0 | 0 | 0 | 0 |
| 0 | 50 | 0 | 190 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 160 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 160 | 0 | 2 | 0 | 0 | 0 | 0 |
| 94 | 0 | 81 | 0 | 0 | 0 | 0 | 0 |
| 0 | 94 | 0 | 81 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 99 | 99 | 99 | 0 |
| 0 | 0 | 0 | 0 | 0 | 43 | 142 | 43 |
| 0 | 0 | 0 | 0 | 198 | 99 | 0 | 198 |
| 0 | 0 | 0 | 0 | 142 | 0 | 99 | 99 |
G:=sub<GL(8,GF(241))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,240,1,0,0,0,0,0,240,0,1,0,0,0,0,1,240,0,0,0,0,0,0,0,239,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,240,0,0,0,0,0,240,0,240,1,0,0,0,0,0,0,240,1,0,0,0,0,0,0,239,1],[240,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,1,0,1,240,0,0,0,0,0,0,1,240,0,0,0,0,0,240,1,0,0,0,0,0,0,0,2,240],[240,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,228,0,62,13,0,0,0,0,0,62,228,13,0,0,0,0,62,228,0,13,0,0,0,0,62,62,62,192],[240,0,50,0,0,0,0,0,0,240,0,50,0,0,0,0,1,0,190,0,0,0,0,0,0,1,0,190,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[160,0,94,0,0,0,0,0,0,160,0,94,0,0,0,0,2,0,81,0,0,0,0,0,0,2,0,81,0,0,0,0,0,0,0,0,99,0,198,142,0,0,0,0,99,43,99,0,0,0,0,0,99,142,0,99,0,0,0,0,0,43,198,99] >;
CSU2(𝔽3)⋊D5 in GAP, Magma, Sage, TeX
{\rm CSU}_2({\mathbb F}_3)\rtimes D_5 % in TeX
G:=Group("CSU(2,3):D5"); // GroupNames label
G:=SmallGroup(480,967);
// by ID
G=gap.SmallGroup(480,967);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,1680,3389,1688,93,1347,2111,3168,172,1272,1909,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=c^3=e^5=f^2=1,b^2=d^2=a^2,b*a*b^-1=f*a*f=d*b*d^-1=a^-1,c*a*c^-1=a*b,d*a*d^-1=f*b*f=a^2*b,a*e=e*a,c*b*c^-1=a,b*e=e*b,d*c*d^-1=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
Export