metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.17D6, D30.43D4, C60.46C23, Dic15.17D4, Dic6.17D10, Dic30.17C22, Q8⋊D5⋊6S3, C3⋊C8.11D10, (Q8×D15)⋊2C2, C3⋊Q16⋊6D5, C6.80(D4×D5), C5⋊Dic12⋊8C2, C10.81(S3×D4), C5⋊2C8.11D6, C5⋊4(D4.D6), Q8.23(S3×D5), (C5×Q8).27D6, C3⋊3(Q16⋊D5), C30.208(C2×D4), C6.D20⋊8C2, (C3×Q8).10D10, D30.5C4⋊7C2, D20⋊S3.1C2, C15⋊23(C8.C22), C20.46(C22×S3), C12.46(C22×D5), (C3×D20).18C22, (C4×D15).14C22, (Q8×C15).16C22, C2.33(D10⋊D6), (C5×Dic6).18C22, C4.46(C2×S3×D5), (C3×Q8⋊D5)⋊8C2, (C5×C3⋊Q16)⋊8C2, (C5×C3⋊C8).16C22, (C3×C5⋊2C8).16C22, SmallGroup(480,598)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.17D6
G = < a,b,c,d | a30=b2=d2=1, c4=a15, bab=a-1, cac-1=dad=a19, cbc-1=dbd=a3b, dcd=c3 >
Subgroups: 700 in 120 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22 [×2], C5, S3, C6, C6, C8 [×2], C2×C4 [×3], D4 [×2], Q8, Q8 [×3], D5 [×2], C10, Dic3 [×3], C12, C12, D6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5 [×2], C20, C20 [×2], D10 [×2], C3⋊C8, C24, Dic6, Dic6 [×2], C4×S3 [×2], C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C3×D5, D15, C30, C8.C22, C5⋊2C8, C40, Dic10 [×2], C4×D5 [×3], D20, D20, C5×Q8, C5×Q8, C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×SD16, D4⋊2S3, S3×Q8, C5×Dic3, Dic15, Dic15, C60, C60, C6×D5, D30, C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q8⋊2D5, D4.D6, C5×C3⋊C8, C3×C5⋊2C8, D5×Dic3, C3⋊D20, C3×D20, C5×Dic6, Dic30, Dic30, C4×D15, C4×D15, Q8×C15, Q16⋊D5, D30.5C4, C6.D20, C5⋊Dic12, C3×Q8⋊D5, C5×C3⋊Q16, D20⋊S3, Q8×D15, D20.17D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8.C22, C22×D5, S3×D4, S3×D5, D4×D5, D4.D6, C2×S3×D5, Q16⋊D5, D10⋊D6, D20.17D6
►On 240 points
Generators in S240
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(51 60)(52 59)(53 58)(54 57)(55 56)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(81 90)(82 89)(83 88)(84 87)(85 86)(91 110)(92 109)(93 108)(94 107)(95 106)(96 105)(97 104)(98 103)(99 102)(100 101)(111 120)(112 119)(113 118)(114 117)(115 116)(121 145)(122 144)(123 143)(124 142)(125 141)(126 140)(127 139)(128 138)(129 137)(130 136)(131 135)(132 134)(146 150)(147 149)(151 165)(152 164)(153 163)(154 162)(155 161)(156 160)(157 159)(166 180)(167 179)(168 178)(169 177)(170 176)(171 175)(172 174)(181 185)(182 184)(186 210)(187 209)(188 208)(189 207)(190 206)(191 205)(192 204)(193 203)(194 202)(195 201)(196 200)(197 199)(211 225)(212 224)(213 223)(214 222)(215 221)(216 220)(217 219)(226 240)(227 239)(228 238)(229 237)(230 236)(231 235)(232 234)
(1 226 56 191 16 211 41 206)(2 215 57 210 17 230 42 195)(3 234 58 199 18 219 43 184)(4 223 59 188 19 238 44 203)(5 212 60 207 20 227 45 192)(6 231 31 196 21 216 46 181)(7 220 32 185 22 235 47 200)(8 239 33 204 23 224 48 189)(9 228 34 193 24 213 49 208)(10 217 35 182 25 232 50 197)(11 236 36 201 26 221 51 186)(12 225 37 190 27 240 52 205)(13 214 38 209 28 229 53 194)(14 233 39 198 29 218 54 183)(15 222 40 187 30 237 55 202)(61 171 106 131 76 156 91 146)(62 160 107 150 77 175 92 135)(63 179 108 139 78 164 93 124)(64 168 109 128 79 153 94 143)(65 157 110 147 80 172 95 132)(66 176 111 136 81 161 96 121)(67 165 112 125 82 180 97 140)(68 154 113 144 83 169 98 129)(69 173 114 133 84 158 99 148)(70 162 115 122 85 177 100 137)(71 151 116 141 86 166 101 126)(72 170 117 130 87 155 102 145)(73 159 118 149 88 174 103 134)(74 178 119 138 89 163 104 123)(75 167 120 127 90 152 105 142)
(1 126)(2 145)(3 134)(4 123)(5 142)(6 131)(7 150)(8 139)(9 128)(10 147)(11 136)(12 125)(13 144)(14 133)(15 122)(16 141)(17 130)(18 149)(19 138)(20 127)(21 146)(22 135)(23 124)(24 143)(25 132)(26 121)(27 140)(28 129)(29 148)(30 137)(31 171)(32 160)(33 179)(34 168)(35 157)(36 176)(37 165)(38 154)(39 173)(40 162)(41 151)(42 170)(43 159)(44 178)(45 167)(46 156)(47 175)(48 164)(49 153)(50 172)(51 161)(52 180)(53 169)(54 158)(55 177)(56 166)(57 155)(58 174)(59 163)(60 152)(61 181)(62 200)(63 189)(64 208)(65 197)(66 186)(67 205)(68 194)(69 183)(70 202)(71 191)(72 210)(73 199)(74 188)(75 207)(76 196)(77 185)(78 204)(79 193)(80 182)(81 201)(82 190)(83 209)(84 198)(85 187)(86 206)(87 195)(88 184)(89 203)(90 192)(91 231)(92 220)(93 239)(94 228)(95 217)(96 236)(97 225)(98 214)(99 233)(100 222)(101 211)(102 230)(103 219)(104 238)(105 227)(106 216)(107 235)(108 224)(109 213)(110 232)(111 221)(112 240)(113 229)(114 218)(115 237)(116 226)(117 215)(118 234)(119 223)(120 212)
G:=sub<Sym(240)| (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(51,60)(52,59)(53,58)(54,57)(55,56)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,90)(82,89)(83,88)(84,87)(85,86)(91,110)(92,109)(93,108)(94,107)(95,106)(96,105)(97,104)(98,103)(99,102)(100,101)(111,120)(112,119)(113,118)(114,117)(115,116)(121,145)(122,144)(123,143)(124,142)(125,141)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(146,150)(147,149)(151,165)(152,164)(153,163)(154,162)(155,161)(156,160)(157,159)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(172,174)(181,185)(182,184)(186,210)(187,209)(188,208)(189,207)(190,206)(191,205)(192,204)(193,203)(194,202)(195,201)(196,200)(197,199)(211,225)(212,224)(213,223)(214,222)(215,221)(216,220)(217,219)(226,240)(227,239)(228,238)(229,237)(230,236)(231,235)(232,234), (1,226,56,191,16,211,41,206)(2,215,57,210,17,230,42,195)(3,234,58,199,18,219,43,184)(4,223,59,188,19,238,44,203)(5,212,60,207,20,227,45,192)(6,231,31,196,21,216,46,181)(7,220,32,185,22,235,47,200)(8,239,33,204,23,224,48,189)(9,228,34,193,24,213,49,208)(10,217,35,182,25,232,50,197)(11,236,36,201,26,221,51,186)(12,225,37,190,27,240,52,205)(13,214,38,209,28,229,53,194)(14,233,39,198,29,218,54,183)(15,222,40,187,30,237,55,202)(61,171,106,131,76,156,91,146)(62,160,107,150,77,175,92,135)(63,179,108,139,78,164,93,124)(64,168,109,128,79,153,94,143)(65,157,110,147,80,172,95,132)(66,176,111,136,81,161,96,121)(67,165,112,125,82,180,97,140)(68,154,113,144,83,169,98,129)(69,173,114,133,84,158,99,148)(70,162,115,122,85,177,100,137)(71,151,116,141,86,166,101,126)(72,170,117,130,87,155,102,145)(73,159,118,149,88,174,103,134)(74,178,119,138,89,163,104,123)(75,167,120,127,90,152,105,142), (1,126)(2,145)(3,134)(4,123)(5,142)(6,131)(7,150)(8,139)(9,128)(10,147)(11,136)(12,125)(13,144)(14,133)(15,122)(16,141)(17,130)(18,149)(19,138)(20,127)(21,146)(22,135)(23,124)(24,143)(25,132)(26,121)(27,140)(28,129)(29,148)(30,137)(31,171)(32,160)(33,179)(34,168)(35,157)(36,176)(37,165)(38,154)(39,173)(40,162)(41,151)(42,170)(43,159)(44,178)(45,167)(46,156)(47,175)(48,164)(49,153)(50,172)(51,161)(52,180)(53,169)(54,158)(55,177)(56,166)(57,155)(58,174)(59,163)(60,152)(61,181)(62,200)(63,189)(64,208)(65,197)(66,186)(67,205)(68,194)(69,183)(70,202)(71,191)(72,210)(73,199)(74,188)(75,207)(76,196)(77,185)(78,204)(79,193)(80,182)(81,201)(82,190)(83,209)(84,198)(85,187)(86,206)(87,195)(88,184)(89,203)(90,192)(91,231)(92,220)(93,239)(94,228)(95,217)(96,236)(97,225)(98,214)(99,233)(100,222)(101,211)(102,230)(103,219)(104,238)(105,227)(106,216)(107,235)(108,224)(109,213)(110,232)(111,221)(112,240)(113,229)(114,218)(115,237)(116,226)(117,215)(118,234)(119,223)(120,212)>;
G:=Group( (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(51,60)(52,59)(53,58)(54,57)(55,56)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,90)(82,89)(83,88)(84,87)(85,86)(91,110)(92,109)(93,108)(94,107)(95,106)(96,105)(97,104)(98,103)(99,102)(100,101)(111,120)(112,119)(113,118)(114,117)(115,116)(121,145)(122,144)(123,143)(124,142)(125,141)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(146,150)(147,149)(151,165)(152,164)(153,163)(154,162)(155,161)(156,160)(157,159)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(172,174)(181,185)(182,184)(186,210)(187,209)(188,208)(189,207)(190,206)(191,205)(192,204)(193,203)(194,202)(195,201)(196,200)(197,199)(211,225)(212,224)(213,223)(214,222)(215,221)(216,220)(217,219)(226,240)(227,239)(228,238)(229,237)(230,236)(231,235)(232,234), (1,226,56,191,16,211,41,206)(2,215,57,210,17,230,42,195)(3,234,58,199,18,219,43,184)(4,223,59,188,19,238,44,203)(5,212,60,207,20,227,45,192)(6,231,31,196,21,216,46,181)(7,220,32,185,22,235,47,200)(8,239,33,204,23,224,48,189)(9,228,34,193,24,213,49,208)(10,217,35,182,25,232,50,197)(11,236,36,201,26,221,51,186)(12,225,37,190,27,240,52,205)(13,214,38,209,28,229,53,194)(14,233,39,198,29,218,54,183)(15,222,40,187,30,237,55,202)(61,171,106,131,76,156,91,146)(62,160,107,150,77,175,92,135)(63,179,108,139,78,164,93,124)(64,168,109,128,79,153,94,143)(65,157,110,147,80,172,95,132)(66,176,111,136,81,161,96,121)(67,165,112,125,82,180,97,140)(68,154,113,144,83,169,98,129)(69,173,114,133,84,158,99,148)(70,162,115,122,85,177,100,137)(71,151,116,141,86,166,101,126)(72,170,117,130,87,155,102,145)(73,159,118,149,88,174,103,134)(74,178,119,138,89,163,104,123)(75,167,120,127,90,152,105,142), (1,126)(2,145)(3,134)(4,123)(5,142)(6,131)(7,150)(8,139)(9,128)(10,147)(11,136)(12,125)(13,144)(14,133)(15,122)(16,141)(17,130)(18,149)(19,138)(20,127)(21,146)(22,135)(23,124)(24,143)(25,132)(26,121)(27,140)(28,129)(29,148)(30,137)(31,171)(32,160)(33,179)(34,168)(35,157)(36,176)(37,165)(38,154)(39,173)(40,162)(41,151)(42,170)(43,159)(44,178)(45,167)(46,156)(47,175)(48,164)(49,153)(50,172)(51,161)(52,180)(53,169)(54,158)(55,177)(56,166)(57,155)(58,174)(59,163)(60,152)(61,181)(62,200)(63,189)(64,208)(65,197)(66,186)(67,205)(68,194)(69,183)(70,202)(71,191)(72,210)(73,199)(74,188)(75,207)(76,196)(77,185)(78,204)(79,193)(80,182)(81,201)(82,190)(83,209)(84,198)(85,187)(86,206)(87,195)(88,184)(89,203)(90,192)(91,231)(92,220)(93,239)(94,228)(95,217)(96,236)(97,225)(98,214)(99,233)(100,222)(101,211)(102,230)(103,219)(104,238)(105,227)(106,216)(107,235)(108,224)(109,213)(110,232)(111,221)(112,240)(113,229)(114,218)(115,237)(116,226)(117,215)(118,234)(119,223)(120,212) );
G=PermutationGroup([(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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(51,60),(52,59),(53,58),(54,57),(55,56),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71),(81,90),(82,89),(83,88),(84,87),(85,86),(91,110),(92,109),(93,108),(94,107),(95,106),(96,105),(97,104),(98,103),(99,102),(100,101),(111,120),(112,119),(113,118),(114,117),(115,116),(121,145),(122,144),(123,143),(124,142),(125,141),(126,140),(127,139),(128,138),(129,137),(130,136),(131,135),(132,134),(146,150),(147,149),(151,165),(152,164),(153,163),(154,162),(155,161),(156,160),(157,159),(166,180),(167,179),(168,178),(169,177),(170,176),(171,175),(172,174),(181,185),(182,184),(186,210),(187,209),(188,208),(189,207),(190,206),(191,205),(192,204),(193,203),(194,202),(195,201),(196,200),(197,199),(211,225),(212,224),(213,223),(214,222),(215,221),(216,220),(217,219),(226,240),(227,239),(228,238),(229,237),(230,236),(231,235),(232,234)], [(1,226,56,191,16,211,41,206),(2,215,57,210,17,230,42,195),(3,234,58,199,18,219,43,184),(4,223,59,188,19,238,44,203),(5,212,60,207,20,227,45,192),(6,231,31,196,21,216,46,181),(7,220,32,185,22,235,47,200),(8,239,33,204,23,224,48,189),(9,228,34,193,24,213,49,208),(10,217,35,182,25,232,50,197),(11,236,36,201,26,221,51,186),(12,225,37,190,27,240,52,205),(13,214,38,209,28,229,53,194),(14,233,39,198,29,218,54,183),(15,222,40,187,30,237,55,202),(61,171,106,131,76,156,91,146),(62,160,107,150,77,175,92,135),(63,179,108,139,78,164,93,124),(64,168,109,128,79,153,94,143),(65,157,110,147,80,172,95,132),(66,176,111,136,81,161,96,121),(67,165,112,125,82,180,97,140),(68,154,113,144,83,169,98,129),(69,173,114,133,84,158,99,148),(70,162,115,122,85,177,100,137),(71,151,116,141,86,166,101,126),(72,170,117,130,87,155,102,145),(73,159,118,149,88,174,103,134),(74,178,119,138,89,163,104,123),(75,167,120,127,90,152,105,142)], [(1,126),(2,145),(3,134),(4,123),(5,142),(6,131),(7,150),(8,139),(9,128),(10,147),(11,136),(12,125),(13,144),(14,133),(15,122),(16,141),(17,130),(18,149),(19,138),(20,127),(21,146),(22,135),(23,124),(24,143),(25,132),(26,121),(27,140),(28,129),(29,148),(30,137),(31,171),(32,160),(33,179),(34,168),(35,157),(36,176),(37,165),(38,154),(39,173),(40,162),(41,151),(42,170),(43,159),(44,178),(45,167),(46,156),(47,175),(48,164),(49,153),(50,172),(51,161),(52,180),(53,169),(54,158),(55,177),(56,166),(57,155),(58,174),(59,163),(60,152),(61,181),(62,200),(63,189),(64,208),(65,197),(66,186),(67,205),(68,194),(69,183),(70,202),(71,191),(72,210),(73,199),(74,188),(75,207),(76,196),(77,185),(78,204),(79,193),(80,182),(81,201),(82,190),(83,209),(84,198),(85,187),(86,206),(87,195),(88,184),(89,203),(90,192),(91,231),(92,220),(93,239),(94,228),(95,217),(96,236),(97,225),(98,214),(99,233),(100,222),(101,211),(102,230),(103,219),(104,238),(105,227),(106,216),(107,235),(108,224),(109,213),(110,232),(111,221),(112,240),(113,229),(114,218),(115,237),(116,226),(117,215),(118,234),(119,223),(120,212)])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 30A | 30B | 40A | 40B | 40C | 40D | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 20 | 30 | 2 | 2 | 4 | 12 | 30 | 60 | 2 | 2 | 2 | 40 | 12 | 20 | 2 | 2 | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 24 | 24 | 20 | 20 | 4 | 4 | 12 | 12 | 12 | 12 | 8 | ··· | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C8.C22 | S3×D4 | S3×D5 | D4×D5 | D4.D6 | C2×S3×D5 | Q16⋊D5 | D10⋊D6 | D20.17D6 |
| kernel | D20.17D6 | D30.5C4 | C6.D20 | C5⋊Dic12 | C3×Q8⋊D5 | C5×C3⋊Q16 | D20⋊S3 | Q8×D15 | Q8⋊D5 | Dic15 | D30 | C3⋊Q16 | C5⋊2C8 | D20 | C5×Q8 | C3⋊C8 | Dic6 | C3×Q8 | C15 | C10 | Q8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D20.17D6 ►in GL8(𝔽241)
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 51 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 240 | 190 | 240 | 190 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 51 | 191 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 0 | 190 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 190 | 51 | 0 | 0 | 0 | 0 |
| 190 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 190 | 51 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 189 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 189 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 51 |
| 0 | 0 | 0 | 0 | 0 | 0 | 52 | 0 |
| 11 | 20 | 31 | 31 | 0 | 0 | 0 | 0 |
| 182 | 230 | 137 | 210 | 0 | 0 | 0 | 0 |
| 210 | 210 | 221 | 230 | 0 | 0 | 0 | 0 |
| 104 | 31 | 45 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 77 | 148 |
| 0 | 0 | 0 | 0 | 0 | 0 | 225 | 71 |
| 0 | 0 | 0 | 0 | 173 | 230 | 169 | 57 |
| 0 | 0 | 0 | 0 | 151 | 79 | 57 | 72 |
| 104 | 238 | 101 | 101 | 0 | 0 | 0 | 0 |
| 236 | 137 | 11 | 140 | 0 | 0 | 0 | 0 |
| 140 | 140 | 3 | 137 | 0 | 0 | 0 | 0 |
| 230 | 101 | 225 | 238 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 202 | 219 | 76 | 96 |
| 0 | 0 | 0 | 0 | 119 | 39 | 172 | 20 |
| 0 | 0 | 0 | 0 | 91 | 238 | 180 | 219 |
| 0 | 0 | 0 | 0 | 85 | 153 | 219 | 61 |
G:=sub<GL(8,GF(241))| [0,0,0,240,0,0,0,0,0,0,1,190,0,0,0,0,0,1,0,240,0,0,0,0,240,51,1,190,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,1,52,0,0,0,0,0,0,0,0,51,240,0,0,0,0,0,0,191,1],[0,0,190,190,0,0,0,0,0,0,240,51,0,0,0,0,190,190,0,0,0,0,0,0,240,51,0,0,0,0,0,0,0,0,0,0,52,1,0,0,0,0,0,0,189,189,0,0,0,0,0,0,0,0,0,52,0,0,0,0,0,0,51,0],[11,182,210,104,0,0,0,0,20,230,210,31,0,0,0,0,31,137,221,45,0,0,0,0,31,210,230,20,0,0,0,0,0,0,0,0,0,0,173,151,0,0,0,0,0,0,230,79,0,0,0,0,77,225,169,57,0,0,0,0,148,71,57,72],[104,236,140,230,0,0,0,0,238,137,140,101,0,0,0,0,101,11,3,225,0,0,0,0,101,140,137,238,0,0,0,0,0,0,0,0,202,119,91,85,0,0,0,0,219,39,238,153,0,0,0,0,76,172,180,219,0,0,0,0,96,20,219,61] >;
D20.17D6 in GAP, Magma, Sage, TeX
D_{20}._{17}D_6 % in TeX
G:=Group("D20.17D6"); // GroupNames label
G:=SmallGroup(480,598);
// by ID
G=gap.SmallGroup(480,598);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,303,100,675,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=d^2=1,c^4=a^15,b*a*b=a^-1,c*a*c^-1=d*a*d=a^19,c*b*c^-1=d*b*d=a^3*b,d*c*d=c^3>;
// generators/relations