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G = D5×C3⋊Q16order 480 = 25·3·5

Direct product of D5 and C3⋊Q16

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D5×C3⋊Q16, C60.31C23, Dic6.25D10, Dic10.11D6, Dic30.8C22, C35(D5×Q16), C157(C2×Q16), C3⋊C8.17D10, (C3×D5)⋊2Q16, (Q8×D5).3S3, C3⋊Dic205C2, (C6×D5).65D4, C15⋊Q165C2, C157Q161C2, (C4×D5).48D6, C6.148(D4×D5), Q8.14(S3×D5), (C5×Q8).18D6, C30.193(C2×D4), (C3×Q8).18D10, (D5×Dic6).1C2, C20.31(C22×S3), C153C8.7C22, (C3×Dic5).16D4, C12.31(C22×D5), (Q8×C15).1C22, D10.41(C3⋊D4), (D5×C12).11C22, (C5×Dic6).8C22, Dic5.14(C3⋊D4), (C3×Dic10).8C22, (D5×C3⋊C8).1C2, C4.31(C2×S3×D5), C52(C2×C3⋊Q16), (C3×Q8×D5).1C2, (C5×C3⋊Q16)⋊1C2, C2.30(D5×C3⋊D4), (C5×C3⋊C8).7C22, C10.51(C2×C3⋊D4), SmallGroup(480,583)

Series: Derived Chief Lower central Upper central

C1C60 — D5×C3⋊Q16
C1C5C15C30C60D5×C12D5×Dic6 — D5×C3⋊Q16
C15C30C60 — D5×C3⋊Q16
C1C2C4Q8

Generators and relations for D5×C3⋊Q16
 G = < a,b,c,d,e | a5=b2=c3=d8=1, e2=d4, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=d-1 >

Subgroups: 556 in 120 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×5], C22, C5, C6, C6 [×2], C8 [×2], C2×C4 [×3], Q8, Q8 [×5], D5 [×2], C10, Dic3 [×2], C12, C12 [×3], C2×C6, C15, C2×C8, Q16 [×4], C2×Q8 [×2], Dic5, Dic5 [×2], C20, C20 [×2], D10, C3⋊C8, C3⋊C8, Dic6, Dic6 [×2], C2×Dic3, C2×C12 [×2], C3×Q8, C3×Q8 [×2], C3×D5 [×2], C30, C2×Q16, C52C8, C40, Dic10, Dic10 [×3], C4×D5, C4×D5 [×2], C5×Q8, C5×Q8, C2×C3⋊C8, C3⋊Q16, C3⋊Q16 [×3], C2×Dic6, C6×Q8, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, C60, C6×D5, C8×D5, Dic20, C5⋊Q16 [×2], C5×Q16, Q8×D5, Q8×D5, C2×C3⋊Q16, C5×C3⋊C8, C153C8, D5×Dic3, C15⋊Q8, C3×Dic10, C3×Dic10, D5×C12, D5×C12, C5×Dic6, Dic30, Q8×C15, D5×Q16, D5×C3⋊C8, C15⋊Q16, C3⋊Dic20, C5×C3⋊Q16, C157Q16, D5×Dic6, C3×Q8×D5, D5×C3⋊Q16
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], Q16 [×2], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C2×Q16, C22×D5, C3⋊Q16 [×2], C2×C3⋊D4, S3×D5, D4×D5, C2×C3⋊Q16, C2×S3×D5, D5×Q16, D5×C3⋊D4, D5×C3⋊Q16

Smallest permutation representation of D5×C3⋊Q16
On 240 points
Generators in S240
(1 193 233 125 76)(2 194 234 126 77)(3 195 235 127 78)(4 196 236 128 79)(5 197 237 121 80)(6 198 238 122 73)(7 199 239 123 74)(8 200 240 124 75)(9 231 91 210 26)(10 232 92 211 27)(11 225 93 212 28)(12 226 94 213 29)(13 227 95 214 30)(14 228 96 215 31)(15 229 89 216 32)(16 230 90 209 25)(17 36 101 150 203)(18 37 102 151 204)(19 38 103 152 205)(20 39 104 145 206)(21 40 97 146 207)(22 33 98 147 208)(23 34 99 148 201)(24 35 100 149 202)(41 180 144 157 112)(42 181 137 158 105)(43 182 138 159 106)(44 183 139 160 107)(45 184 140 153 108)(46 177 141 154 109)(47 178 142 155 110)(48 179 143 156 111)(49 188 220 167 118)(50 189 221 168 119)(51 190 222 161 120)(52 191 223 162 113)(53 192 224 163 114)(54 185 217 164 115)(55 186 218 165 116)(56 187 219 166 117)(57 176 135 84 65)(58 169 136 85 66)(59 170 129 86 67)(60 171 130 87 68)(61 172 131 88 69)(62 173 132 81 70)(63 174 133 82 71)(64 175 134 83 72)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 73)(7 74)(8 75)(9 210)(10 211)(11 212)(12 213)(13 214)(14 215)(15 216)(16 209)(33 208)(34 201)(35 202)(36 203)(37 204)(38 205)(39 206)(40 207)(41 180)(42 181)(43 182)(44 183)(45 184)(46 177)(47 178)(48 179)(49 167)(50 168)(51 161)(52 162)(53 163)(54 164)(55 165)(56 166)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)(63 71)(64 72)(81 173)(82 174)(83 175)(84 176)(85 169)(86 170)(87 171)(88 172)(89 229)(90 230)(91 231)(92 232)(93 225)(94 226)(95 227)(96 228)(97 146)(98 147)(99 148)(100 149)(101 150)(102 151)(103 152)(104 145)(105 137)(106 138)(107 139)(108 140)(109 141)(110 142)(111 143)(112 144)(121 197)(122 198)(123 199)(124 200)(125 193)(126 194)(127 195)(128 196)(185 217)(186 218)(187 219)(188 220)(189 221)(190 222)(191 223)(192 224)
(1 180 95)(2 96 181)(3 182 89)(4 90 183)(5 184 91)(6 92 177)(7 178 93)(8 94 179)(9 121 108)(10 109 122)(11 123 110)(12 111 124)(13 125 112)(14 105 126)(15 127 106)(16 107 128)(17 135 118)(18 119 136)(19 129 120)(20 113 130)(21 131 114)(22 115 132)(23 133 116)(24 117 134)(25 160 236)(26 237 153)(27 154 238)(28 239 155)(29 156 240)(30 233 157)(31 158 234)(32 235 159)(33 54 81)(34 82 55)(35 56 83)(36 84 49)(37 50 85)(38 86 51)(39 52 87)(40 88 53)(41 227 76)(42 77 228)(43 229 78)(44 79 230)(45 231 80)(46 73 232)(47 225 74)(48 75 226)(57 220 150)(58 151 221)(59 222 152)(60 145 223)(61 224 146)(62 147 217)(63 218 148)(64 149 219)(65 188 101)(66 102 189)(67 190 103)(68 104 191)(69 192 97)(70 98 185)(71 186 99)(72 100 187)(137 194 215)(138 216 195)(139 196 209)(140 210 197)(141 198 211)(142 212 199)(143 200 213)(144 214 193)(161 205 170)(162 171 206)(163 207 172)(164 173 208)(165 201 174)(166 175 202)(167 203 176)(168 169 204)
(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)(161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176)(177 178 179 180 181 182 183 184)(185 186 187 188 189 190 191 192)(193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208)(209 210 211 212 213 214 215 216)(217 218 219 220 221 222 223 224)(225 226 227 228 229 230 231 232)(233 234 235 236 237 238 239 240)
(1 222 5 218)(2 221 6 217)(3 220 7 224)(4 219 8 223)(9 82 13 86)(10 81 14 85)(11 88 15 84)(12 87 16 83)(17 155 21 159)(18 154 22 158)(19 153 23 157)(20 160 24 156)(25 134 29 130)(26 133 30 129)(27 132 31 136)(28 131 32 135)(33 105 37 109)(34 112 38 108)(35 111 39 107)(36 110 40 106)(41 103 45 99)(42 102 46 98)(43 101 47 97)(44 100 48 104)(49 123 53 127)(50 122 54 126)(51 121 55 125)(52 128 56 124)(57 93 61 89)(58 92 62 96)(59 91 63 95)(60 90 64 94)(65 225 69 229)(66 232 70 228)(67 231 71 227)(68 230 72 226)(73 185 77 189)(74 192 78 188)(75 191 79 187)(76 190 80 186)(113 236 117 240)(114 235 118 239)(115 234 119 238)(116 233 120 237)(137 204 141 208)(138 203 142 207)(139 202 143 206)(140 201 144 205)(145 183 149 179)(146 182 150 178)(147 181 151 177)(148 180 152 184)(161 197 165 193)(162 196 166 200)(163 195 167 199)(164 194 168 198)(169 211 173 215)(170 210 174 214)(171 209 175 213)(172 216 176 212)

G:=sub<Sym(240)| (1,193,233,125,76)(2,194,234,126,77)(3,195,235,127,78)(4,196,236,128,79)(5,197,237,121,80)(6,198,238,122,73)(7,199,239,123,74)(8,200,240,124,75)(9,231,91,210,26)(10,232,92,211,27)(11,225,93,212,28)(12,226,94,213,29)(13,227,95,214,30)(14,228,96,215,31)(15,229,89,216,32)(16,230,90,209,25)(17,36,101,150,203)(18,37,102,151,204)(19,38,103,152,205)(20,39,104,145,206)(21,40,97,146,207)(22,33,98,147,208)(23,34,99,148,201)(24,35,100,149,202)(41,180,144,157,112)(42,181,137,158,105)(43,182,138,159,106)(44,183,139,160,107)(45,184,140,153,108)(46,177,141,154,109)(47,178,142,155,110)(48,179,143,156,111)(49,188,220,167,118)(50,189,221,168,119)(51,190,222,161,120)(52,191,223,162,113)(53,192,224,163,114)(54,185,217,164,115)(55,186,218,165,116)(56,187,219,166,117)(57,176,135,84,65)(58,169,136,85,66)(59,170,129,86,67)(60,171,130,87,68)(61,172,131,88,69)(62,173,132,81,70)(63,174,133,82,71)(64,175,134,83,72), (1,76)(2,77)(3,78)(4,79)(5,80)(6,73)(7,74)(8,75)(9,210)(10,211)(11,212)(12,213)(13,214)(14,215)(15,216)(16,209)(33,208)(34,201)(35,202)(36,203)(37,204)(38,205)(39,206)(40,207)(41,180)(42,181)(43,182)(44,183)(45,184)(46,177)(47,178)(48,179)(49,167)(50,168)(51,161)(52,162)(53,163)(54,164)(55,165)(56,166)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,71)(64,72)(81,173)(82,174)(83,175)(84,176)(85,169)(86,170)(87,171)(88,172)(89,229)(90,230)(91,231)(92,232)(93,225)(94,226)(95,227)(96,228)(97,146)(98,147)(99,148)(100,149)(101,150)(102,151)(103,152)(104,145)(105,137)(106,138)(107,139)(108,140)(109,141)(110,142)(111,143)(112,144)(121,197)(122,198)(123,199)(124,200)(125,193)(126,194)(127,195)(128,196)(185,217)(186,218)(187,219)(188,220)(189,221)(190,222)(191,223)(192,224), (1,180,95)(2,96,181)(3,182,89)(4,90,183)(5,184,91)(6,92,177)(7,178,93)(8,94,179)(9,121,108)(10,109,122)(11,123,110)(12,111,124)(13,125,112)(14,105,126)(15,127,106)(16,107,128)(17,135,118)(18,119,136)(19,129,120)(20,113,130)(21,131,114)(22,115,132)(23,133,116)(24,117,134)(25,160,236)(26,237,153)(27,154,238)(28,239,155)(29,156,240)(30,233,157)(31,158,234)(32,235,159)(33,54,81)(34,82,55)(35,56,83)(36,84,49)(37,50,85)(38,86,51)(39,52,87)(40,88,53)(41,227,76)(42,77,228)(43,229,78)(44,79,230)(45,231,80)(46,73,232)(47,225,74)(48,75,226)(57,220,150)(58,151,221)(59,222,152)(60,145,223)(61,224,146)(62,147,217)(63,218,148)(64,149,219)(65,188,101)(66,102,189)(67,190,103)(68,104,191)(69,192,97)(70,98,185)(71,186,99)(72,100,187)(137,194,215)(138,216,195)(139,196,209)(140,210,197)(141,198,211)(142,212,199)(143,200,213)(144,214,193)(161,205,170)(162,171,206)(163,207,172)(164,173,208)(165,201,174)(166,175,202)(167,203,176)(168,169,204), (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)(161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176)(177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224)(225,226,227,228,229,230,231,232)(233,234,235,236,237,238,239,240), (1,222,5,218)(2,221,6,217)(3,220,7,224)(4,219,8,223)(9,82,13,86)(10,81,14,85)(11,88,15,84)(12,87,16,83)(17,155,21,159)(18,154,22,158)(19,153,23,157)(20,160,24,156)(25,134,29,130)(26,133,30,129)(27,132,31,136)(28,131,32,135)(33,105,37,109)(34,112,38,108)(35,111,39,107)(36,110,40,106)(41,103,45,99)(42,102,46,98)(43,101,47,97)(44,100,48,104)(49,123,53,127)(50,122,54,126)(51,121,55,125)(52,128,56,124)(57,93,61,89)(58,92,62,96)(59,91,63,95)(60,90,64,94)(65,225,69,229)(66,232,70,228)(67,231,71,227)(68,230,72,226)(73,185,77,189)(74,192,78,188)(75,191,79,187)(76,190,80,186)(113,236,117,240)(114,235,118,239)(115,234,119,238)(116,233,120,237)(137,204,141,208)(138,203,142,207)(139,202,143,206)(140,201,144,205)(145,183,149,179)(146,182,150,178)(147,181,151,177)(148,180,152,184)(161,197,165,193)(162,196,166,200)(163,195,167,199)(164,194,168,198)(169,211,173,215)(170,210,174,214)(171,209,175,213)(172,216,176,212)>;

G:=Group( (1,193,233,125,76)(2,194,234,126,77)(3,195,235,127,78)(4,196,236,128,79)(5,197,237,121,80)(6,198,238,122,73)(7,199,239,123,74)(8,200,240,124,75)(9,231,91,210,26)(10,232,92,211,27)(11,225,93,212,28)(12,226,94,213,29)(13,227,95,214,30)(14,228,96,215,31)(15,229,89,216,32)(16,230,90,209,25)(17,36,101,150,203)(18,37,102,151,204)(19,38,103,152,205)(20,39,104,145,206)(21,40,97,146,207)(22,33,98,147,208)(23,34,99,148,201)(24,35,100,149,202)(41,180,144,157,112)(42,181,137,158,105)(43,182,138,159,106)(44,183,139,160,107)(45,184,140,153,108)(46,177,141,154,109)(47,178,142,155,110)(48,179,143,156,111)(49,188,220,167,118)(50,189,221,168,119)(51,190,222,161,120)(52,191,223,162,113)(53,192,224,163,114)(54,185,217,164,115)(55,186,218,165,116)(56,187,219,166,117)(57,176,135,84,65)(58,169,136,85,66)(59,170,129,86,67)(60,171,130,87,68)(61,172,131,88,69)(62,173,132,81,70)(63,174,133,82,71)(64,175,134,83,72), (1,76)(2,77)(3,78)(4,79)(5,80)(6,73)(7,74)(8,75)(9,210)(10,211)(11,212)(12,213)(13,214)(14,215)(15,216)(16,209)(33,208)(34,201)(35,202)(36,203)(37,204)(38,205)(39,206)(40,207)(41,180)(42,181)(43,182)(44,183)(45,184)(46,177)(47,178)(48,179)(49,167)(50,168)(51,161)(52,162)(53,163)(54,164)(55,165)(56,166)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,71)(64,72)(81,173)(82,174)(83,175)(84,176)(85,169)(86,170)(87,171)(88,172)(89,229)(90,230)(91,231)(92,232)(93,225)(94,226)(95,227)(96,228)(97,146)(98,147)(99,148)(100,149)(101,150)(102,151)(103,152)(104,145)(105,137)(106,138)(107,139)(108,140)(109,141)(110,142)(111,143)(112,144)(121,197)(122,198)(123,199)(124,200)(125,193)(126,194)(127,195)(128,196)(185,217)(186,218)(187,219)(188,220)(189,221)(190,222)(191,223)(192,224), (1,180,95)(2,96,181)(3,182,89)(4,90,183)(5,184,91)(6,92,177)(7,178,93)(8,94,179)(9,121,108)(10,109,122)(11,123,110)(12,111,124)(13,125,112)(14,105,126)(15,127,106)(16,107,128)(17,135,118)(18,119,136)(19,129,120)(20,113,130)(21,131,114)(22,115,132)(23,133,116)(24,117,134)(25,160,236)(26,237,153)(27,154,238)(28,239,155)(29,156,240)(30,233,157)(31,158,234)(32,235,159)(33,54,81)(34,82,55)(35,56,83)(36,84,49)(37,50,85)(38,86,51)(39,52,87)(40,88,53)(41,227,76)(42,77,228)(43,229,78)(44,79,230)(45,231,80)(46,73,232)(47,225,74)(48,75,226)(57,220,150)(58,151,221)(59,222,152)(60,145,223)(61,224,146)(62,147,217)(63,218,148)(64,149,219)(65,188,101)(66,102,189)(67,190,103)(68,104,191)(69,192,97)(70,98,185)(71,186,99)(72,100,187)(137,194,215)(138,216,195)(139,196,209)(140,210,197)(141,198,211)(142,212,199)(143,200,213)(144,214,193)(161,205,170)(162,171,206)(163,207,172)(164,173,208)(165,201,174)(166,175,202)(167,203,176)(168,169,204), (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)(161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176)(177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224)(225,226,227,228,229,230,231,232)(233,234,235,236,237,238,239,240), (1,222,5,218)(2,221,6,217)(3,220,7,224)(4,219,8,223)(9,82,13,86)(10,81,14,85)(11,88,15,84)(12,87,16,83)(17,155,21,159)(18,154,22,158)(19,153,23,157)(20,160,24,156)(25,134,29,130)(26,133,30,129)(27,132,31,136)(28,131,32,135)(33,105,37,109)(34,112,38,108)(35,111,39,107)(36,110,40,106)(41,103,45,99)(42,102,46,98)(43,101,47,97)(44,100,48,104)(49,123,53,127)(50,122,54,126)(51,121,55,125)(52,128,56,124)(57,93,61,89)(58,92,62,96)(59,91,63,95)(60,90,64,94)(65,225,69,229)(66,232,70,228)(67,231,71,227)(68,230,72,226)(73,185,77,189)(74,192,78,188)(75,191,79,187)(76,190,80,186)(113,236,117,240)(114,235,118,239)(115,234,119,238)(116,233,120,237)(137,204,141,208)(138,203,142,207)(139,202,143,206)(140,201,144,205)(145,183,149,179)(146,182,150,178)(147,181,151,177)(148,180,152,184)(161,197,165,193)(162,196,166,200)(163,195,167,199)(164,194,168,198)(169,211,173,215)(170,210,174,214)(171,209,175,213)(172,216,176,212) );

G=PermutationGroup([(1,193,233,125,76),(2,194,234,126,77),(3,195,235,127,78),(4,196,236,128,79),(5,197,237,121,80),(6,198,238,122,73),(7,199,239,123,74),(8,200,240,124,75),(9,231,91,210,26),(10,232,92,211,27),(11,225,93,212,28),(12,226,94,213,29),(13,227,95,214,30),(14,228,96,215,31),(15,229,89,216,32),(16,230,90,209,25),(17,36,101,150,203),(18,37,102,151,204),(19,38,103,152,205),(20,39,104,145,206),(21,40,97,146,207),(22,33,98,147,208),(23,34,99,148,201),(24,35,100,149,202),(41,180,144,157,112),(42,181,137,158,105),(43,182,138,159,106),(44,183,139,160,107),(45,184,140,153,108),(46,177,141,154,109),(47,178,142,155,110),(48,179,143,156,111),(49,188,220,167,118),(50,189,221,168,119),(51,190,222,161,120),(52,191,223,162,113),(53,192,224,163,114),(54,185,217,164,115),(55,186,218,165,116),(56,187,219,166,117),(57,176,135,84,65),(58,169,136,85,66),(59,170,129,86,67),(60,171,130,87,68),(61,172,131,88,69),(62,173,132,81,70),(63,174,133,82,71),(64,175,134,83,72)], [(1,76),(2,77),(3,78),(4,79),(5,80),(6,73),(7,74),(8,75),(9,210),(10,211),(11,212),(12,213),(13,214),(14,215),(15,216),(16,209),(33,208),(34,201),(35,202),(36,203),(37,204),(38,205),(39,206),(40,207),(41,180),(42,181),(43,182),(44,183),(45,184),(46,177),(47,178),(48,179),(49,167),(50,168),(51,161),(52,162),(53,163),(54,164),(55,165),(56,166),(57,65),(58,66),(59,67),(60,68),(61,69),(62,70),(63,71),(64,72),(81,173),(82,174),(83,175),(84,176),(85,169),(86,170),(87,171),(88,172),(89,229),(90,230),(91,231),(92,232),(93,225),(94,226),(95,227),(96,228),(97,146),(98,147),(99,148),(100,149),(101,150),(102,151),(103,152),(104,145),(105,137),(106,138),(107,139),(108,140),(109,141),(110,142),(111,143),(112,144),(121,197),(122,198),(123,199),(124,200),(125,193),(126,194),(127,195),(128,196),(185,217),(186,218),(187,219),(188,220),(189,221),(190,222),(191,223),(192,224)], [(1,180,95),(2,96,181),(3,182,89),(4,90,183),(5,184,91),(6,92,177),(7,178,93),(8,94,179),(9,121,108),(10,109,122),(11,123,110),(12,111,124),(13,125,112),(14,105,126),(15,127,106),(16,107,128),(17,135,118),(18,119,136),(19,129,120),(20,113,130),(21,131,114),(22,115,132),(23,133,116),(24,117,134),(25,160,236),(26,237,153),(27,154,238),(28,239,155),(29,156,240),(30,233,157),(31,158,234),(32,235,159),(33,54,81),(34,82,55),(35,56,83),(36,84,49),(37,50,85),(38,86,51),(39,52,87),(40,88,53),(41,227,76),(42,77,228),(43,229,78),(44,79,230),(45,231,80),(46,73,232),(47,225,74),(48,75,226),(57,220,150),(58,151,221),(59,222,152),(60,145,223),(61,224,146),(62,147,217),(63,218,148),(64,149,219),(65,188,101),(66,102,189),(67,190,103),(68,104,191),(69,192,97),(70,98,185),(71,186,99),(72,100,187),(137,194,215),(138,216,195),(139,196,209),(140,210,197),(141,198,211),(142,212,199),(143,200,213),(144,214,193),(161,205,170),(162,171,206),(163,207,172),(164,173,208),(165,201,174),(166,175,202),(167,203,176),(168,169,204)], [(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),(161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176),(177,178,179,180,181,182,183,184),(185,186,187,188,189,190,191,192),(193,194,195,196,197,198,199,200),(201,202,203,204,205,206,207,208),(209,210,211,212,213,214,215,216),(217,218,219,220,221,222,223,224),(225,226,227,228,229,230,231,232),(233,234,235,236,237,238,239,240)], [(1,222,5,218),(2,221,6,217),(3,220,7,224),(4,219,8,223),(9,82,13,86),(10,81,14,85),(11,88,15,84),(12,87,16,83),(17,155,21,159),(18,154,22,158),(19,153,23,157),(20,160,24,156),(25,134,29,130),(26,133,30,129),(27,132,31,136),(28,131,32,135),(33,105,37,109),(34,112,38,108),(35,111,39,107),(36,110,40,106),(41,103,45,99),(42,102,46,98),(43,101,47,97),(44,100,48,104),(49,123,53,127),(50,122,54,126),(51,121,55,125),(52,128,56,124),(57,93,61,89),(58,92,62,96),(59,91,63,95),(60,90,64,94),(65,225,69,229),(66,232,70,228),(67,231,71,227),(68,230,72,226),(73,185,77,189),(74,192,78,188),(75,191,79,187),(76,190,80,186),(113,236,117,240),(114,235,118,239),(115,234,119,238),(116,233,120,237),(137,204,141,208),(138,203,142,207),(139,202,143,206),(140,201,144,205),(145,183,149,179),(146,182,150,178),(147,181,151,177),(148,180,152,184),(161,197,165,193),(162,196,166,200),(163,195,167,199),(164,194,168,198),(169,211,173,215),(170,210,174,214),(171,209,175,213),(172,216,176,212)])

48 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D10A10B12A12B12C12D12E12F15A15B20A20B20C20D20E20F30A30B40A40B40C40D60A···60F
order122234444445566688881010121212121212151520202020202030304040404060···60
size115522410122060222101066303022444202020444488242444121212128···8

48 irreducible representations

dim1111111122222222222224444448
type+++++++++++++++-+++-+++--
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6Q16D10D10D10C3⋊D4C3⋊D4C3⋊Q16S3×D5D4×D5C2×S3×D5D5×Q16D5×C3⋊D4D5×C3⋊Q16
kernelD5×C3⋊Q16D5×C3⋊C8C15⋊Q16C3⋊Dic20C5×C3⋊Q16C157Q16D5×Dic6C3×Q8×D5Q8×D5C3×Dic5C6×D5C3⋊Q16Dic10C4×D5C5×Q8C3×D5C3⋊C8Dic6C3×Q8Dic5D10D5Q8C6C4C3C2C1
# reps1111111111121114222222222442

Matrix representation of D5×C3⋊Q16 in GL6(𝔽241)

100000
010000
00189100
00240000
000010
000001
,
100000
010000
00118900
00024000
000010
000001
,
02400000
12400000
001000
000100
000010
000001
,
1661900000
115750000
00240000
00024000
00001111
000023011
,
100000
010000
001000
000100
0000222222
000022219

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,189,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,189,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[166,115,0,0,0,0,190,75,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,11,230,0,0,0,0,11,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,222,222,0,0,0,0,222,19] >;

D5×C3⋊Q16 in GAP, Magma, Sage, TeX

D_5\times C_3\rtimes Q_{16}
% in TeX

G:=Group("D5xC3:Q16");
// GroupNames label

G:=SmallGroup(480,583);
// by ID

G=gap.SmallGroup(480,583);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,135,100,346,185,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^3=d^8=1,e^2=d^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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