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, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, D10, C3⋊C8, C24, Dic6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C3×D5, D15, C30, C8.C22, C5⋊2C8, C40, Dic10, C4×D5, 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, C22, S3, D4, C23, D5, D6, C2×D4, D10, 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 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)(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)(151 175)(152 174)(153 173)(154 172)(155 171)(156 170)(157 169)(158 168)(159 167)(160 166)(161 165)(162 164)(176 180)(177 179)(181 195)(182 194)(183 193)(184 192)(185 191)(186 190)(187 189)(196 210)(197 209)(198 208)(199 207)(200 206)(201 205)(202 204)(211 215)(212 214)(216 240)(217 239)(218 238)(219 237)(220 236)(221 235)(222 234)(223 233)(224 232)(225 231)(226 230)(227 229)
(1 221 41 196 16 236 56 181)(2 240 42 185 17 225 57 200)(3 229 43 204 18 214 58 189)(4 218 44 193 19 233 59 208)(5 237 45 182 20 222 60 197)(6 226 46 201 21 211 31 186)(7 215 47 190 22 230 32 205)(8 234 48 209 23 219 33 194)(9 223 49 198 24 238 34 183)(10 212 50 187 25 227 35 202)(11 231 51 206 26 216 36 191)(12 220 52 195 27 235 37 210)(13 239 53 184 28 224 38 199)(14 228 54 203 29 213 39 188)(15 217 55 192 30 232 40 207)(61 156 101 126 76 171 116 141)(62 175 102 145 77 160 117 130)(63 164 103 134 78 179 118 149)(64 153 104 123 79 168 119 138)(65 172 105 142 80 157 120 127)(66 161 106 131 81 176 91 146)(67 180 107 150 82 165 92 135)(68 169 108 139 83 154 93 124)(69 158 109 128 84 173 94 143)(70 177 110 147 85 162 95 132)(71 166 111 136 86 151 96 121)(72 155 112 125 87 170 97 140)(73 174 113 144 88 159 98 129)(74 163 114 133 89 178 99 148)(75 152 115 122 90 167 100 137)
(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)(31 176)(32 165)(33 154)(34 173)(35 162)(36 151)(37 170)(38 159)(39 178)(40 167)(41 156)(42 175)(43 164)(44 153)(45 172)(46 161)(47 180)(48 169)(49 158)(50 177)(51 166)(52 155)(53 174)(54 163)(55 152)(56 171)(57 160)(58 179)(59 168)(60 157)(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)(91 226)(92 215)(93 234)(94 223)(95 212)(96 231)(97 220)(98 239)(99 228)(100 217)(101 236)(102 225)(103 214)(104 233)(105 222)(106 211)(107 230)(108 219)(109 238)(110 227)(111 216)(112 235)(113 224)(114 213)(115 232)(116 221)(117 240)(118 229)(119 218)(120 237)
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)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(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)(151,175)(152,174)(153,173)(154,172)(155,171)(156,170)(157,169)(158,168)(159,167)(160,166)(161,165)(162,164)(176,180)(177,179)(181,195)(182,194)(183,193)(184,192)(185,191)(186,190)(187,189)(196,210)(197,209)(198,208)(199,207)(200,206)(201,205)(202,204)(211,215)(212,214)(216,240)(217,239)(218,238)(219,237)(220,236)(221,235)(222,234)(223,233)(224,232)(225,231)(226,230)(227,229), (1,221,41,196,16,236,56,181)(2,240,42,185,17,225,57,200)(3,229,43,204,18,214,58,189)(4,218,44,193,19,233,59,208)(5,237,45,182,20,222,60,197)(6,226,46,201,21,211,31,186)(7,215,47,190,22,230,32,205)(8,234,48,209,23,219,33,194)(9,223,49,198,24,238,34,183)(10,212,50,187,25,227,35,202)(11,231,51,206,26,216,36,191)(12,220,52,195,27,235,37,210)(13,239,53,184,28,224,38,199)(14,228,54,203,29,213,39,188)(15,217,55,192,30,232,40,207)(61,156,101,126,76,171,116,141)(62,175,102,145,77,160,117,130)(63,164,103,134,78,179,118,149)(64,153,104,123,79,168,119,138)(65,172,105,142,80,157,120,127)(66,161,106,131,81,176,91,146)(67,180,107,150,82,165,92,135)(68,169,108,139,83,154,93,124)(69,158,109,128,84,173,94,143)(70,177,110,147,85,162,95,132)(71,166,111,136,86,151,96,121)(72,155,112,125,87,170,97,140)(73,174,113,144,88,159,98,129)(74,163,114,133,89,178,99,148)(75,152,115,122,90,167,100,137), (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)(31,176)(32,165)(33,154)(34,173)(35,162)(36,151)(37,170)(38,159)(39,178)(40,167)(41,156)(42,175)(43,164)(44,153)(45,172)(46,161)(47,180)(48,169)(49,158)(50,177)(51,166)(52,155)(53,174)(54,163)(55,152)(56,171)(57,160)(58,179)(59,168)(60,157)(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)(91,226)(92,215)(93,234)(94,223)(95,212)(96,231)(97,220)(98,239)(99,228)(100,217)(101,236)(102,225)(103,214)(104,233)(105,222)(106,211)(107,230)(108,219)(109,238)(110,227)(111,216)(112,235)(113,224)(114,213)(115,232)(116,221)(117,240)(118,229)(119,218)(120,237)>;
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)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(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)(151,175)(152,174)(153,173)(154,172)(155,171)(156,170)(157,169)(158,168)(159,167)(160,166)(161,165)(162,164)(176,180)(177,179)(181,195)(182,194)(183,193)(184,192)(185,191)(186,190)(187,189)(196,210)(197,209)(198,208)(199,207)(200,206)(201,205)(202,204)(211,215)(212,214)(216,240)(217,239)(218,238)(219,237)(220,236)(221,235)(222,234)(223,233)(224,232)(225,231)(226,230)(227,229), (1,221,41,196,16,236,56,181)(2,240,42,185,17,225,57,200)(3,229,43,204,18,214,58,189)(4,218,44,193,19,233,59,208)(5,237,45,182,20,222,60,197)(6,226,46,201,21,211,31,186)(7,215,47,190,22,230,32,205)(8,234,48,209,23,219,33,194)(9,223,49,198,24,238,34,183)(10,212,50,187,25,227,35,202)(11,231,51,206,26,216,36,191)(12,220,52,195,27,235,37,210)(13,239,53,184,28,224,38,199)(14,228,54,203,29,213,39,188)(15,217,55,192,30,232,40,207)(61,156,101,126,76,171,116,141)(62,175,102,145,77,160,117,130)(63,164,103,134,78,179,118,149)(64,153,104,123,79,168,119,138)(65,172,105,142,80,157,120,127)(66,161,106,131,81,176,91,146)(67,180,107,150,82,165,92,135)(68,169,108,139,83,154,93,124)(69,158,109,128,84,173,94,143)(70,177,110,147,85,162,95,132)(71,166,111,136,86,151,96,121)(72,155,112,125,87,170,97,140)(73,174,113,144,88,159,98,129)(74,163,114,133,89,178,99,148)(75,152,115,122,90,167,100,137), (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)(31,176)(32,165)(33,154)(34,173)(35,162)(36,151)(37,170)(38,159)(39,178)(40,167)(41,156)(42,175)(43,164)(44,153)(45,172)(46,161)(47,180)(48,169)(49,158)(50,177)(51,166)(52,155)(53,174)(54,163)(55,152)(56,171)(57,160)(58,179)(59,168)(60,157)(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)(91,226)(92,215)(93,234)(94,223)(95,212)(96,231)(97,220)(98,239)(99,228)(100,217)(101,236)(102,225)(103,214)(104,233)(105,222)(106,211)(107,230)(108,219)(109,238)(110,227)(111,216)(112,235)(113,224)(114,213)(115,232)(116,221)(117,240)(118,229)(119,218)(120,237) );
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),(61,90),(62,89),(63,88),(64,87),(65,86),(66,85),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79),(73,78),(74,77),(75,76),(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),(151,175),(152,174),(153,173),(154,172),(155,171),(156,170),(157,169),(158,168),(159,167),(160,166),(161,165),(162,164),(176,180),(177,179),(181,195),(182,194),(183,193),(184,192),(185,191),(186,190),(187,189),(196,210),(197,209),(198,208),(199,207),(200,206),(201,205),(202,204),(211,215),(212,214),(216,240),(217,239),(218,238),(219,237),(220,236),(221,235),(222,234),(223,233),(224,232),(225,231),(226,230),(227,229)], [(1,221,41,196,16,236,56,181),(2,240,42,185,17,225,57,200),(3,229,43,204,18,214,58,189),(4,218,44,193,19,233,59,208),(5,237,45,182,20,222,60,197),(6,226,46,201,21,211,31,186),(7,215,47,190,22,230,32,205),(8,234,48,209,23,219,33,194),(9,223,49,198,24,238,34,183),(10,212,50,187,25,227,35,202),(11,231,51,206,26,216,36,191),(12,220,52,195,27,235,37,210),(13,239,53,184,28,224,38,199),(14,228,54,203,29,213,39,188),(15,217,55,192,30,232,40,207),(61,156,101,126,76,171,116,141),(62,175,102,145,77,160,117,130),(63,164,103,134,78,179,118,149),(64,153,104,123,79,168,119,138),(65,172,105,142,80,157,120,127),(66,161,106,131,81,176,91,146),(67,180,107,150,82,165,92,135),(68,169,108,139,83,154,93,124),(69,158,109,128,84,173,94,143),(70,177,110,147,85,162,95,132),(71,166,111,136,86,151,96,121),(72,155,112,125,87,170,97,140),(73,174,113,144,88,159,98,129),(74,163,114,133,89,178,99,148),(75,152,115,122,90,167,100,137)], [(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),(31,176),(32,165),(33,154),(34,173),(35,162),(36,151),(37,170),(38,159),(39,178),(40,167),(41,156),(42,175),(43,164),(44,153),(45,172),(46,161),(47,180),(48,169),(49,158),(50,177),(51,166),(52,155),(53,174),(54,163),(55,152),(56,171),(57,160),(58,179),(59,168),(60,157),(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),(91,226),(92,215),(93,234),(94,223),(95,212),(96,231),(97,220),(98,239),(99,228),(100,217),(101,236),(102,225),(103,214),(104,233),(105,222),(106,211),(107,230),(108,219),(109,238),(110,227),(111,216),(112,235),(113,224),(114,213),(115,232),(116,221),(117,240),(118,229),(119,218),(120,237)]])
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