metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.14D6, D12.28D10, C60.38C23, Dic30.14C22, C15⋊D8⋊7C2, C3⋊C8.19D10, C15⋊19(C4○D8), Q8⋊2D5⋊5S3, Q8⋊2S3⋊7D5, (C4×D5).49D6, C15⋊7Q16⋊6C2, (C6×D5).14D4, C6.151(D4×D5), (C5×Q8).23D6, Q8.17(S3×D5), D12⋊5D5⋊3C2, C30.200(C2×D4), C6.D20⋊7C2, C5⋊4(Q8.13D6), (C3×Q8).21D10, C3⋊7(SD16⋊3D5), C20.38(C22×S3), (C3×Dic5).72D4, C12.38(C22×D5), (Q8×C15).8C22, D10.11(C3⋊D4), C15⋊3C8.12C22, (D5×C12).14C22, (C5×D12).14C22, (C3×D20).13C22, Dic5.42(C3⋊D4), (D5×C3⋊C8)⋊6C2, C4.38(C2×S3×D5), C2.33(D5×C3⋊D4), (C5×Q8⋊2S3)⋊6C2, (C3×Q8⋊2D5)⋊2C2, C10.54(C2×C3⋊D4), (C5×C3⋊C8).12C22, SmallGroup(480,590)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.14D6
G = < a,b,c,d | a20=b2=c6=1, d2=a10, bab=dad-1=a-1, cac-1=a9, cbc-1=a18b, dbd-1=a13b, dcd-1=a10c-1 >
Subgroups: 636 in 124 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3, C6, C6 [×2], C8 [×2], C2×C4 [×3], D4 [×4], Q8, Q8, D5 [×2], C10, C10, Dic3, C12, C12 [×2], D6, C2×C6 [×2], C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, Dic5, C20, C20, D10, D10, C2×C10, C3⋊C8, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12 [×2], C3×D4 [×2], C3×Q8, C5×S3, C3×D5 [×2], C30, C4○D8, C5⋊2C8, C40, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C4○D12, C3×C4○D4, C3×Dic5, Dic15, C60, C60, C6×D5, C6×D5, S3×C10, C8×D5, C40⋊C2, D4⋊D5, C5⋊Q16, C5×SD16, D4⋊2D5, Q8⋊2D5, Q8.13D6, C5×C3⋊C8, C15⋊3C8, S3×Dic5, C15⋊D4, D5×C12, D5×C12, C3×D20, C3×D20, C5×D12, Dic30, Q8×C15, SD16⋊3D5, D5×C3⋊C8, C15⋊D8, C6.D20, C5×Q8⋊2S3, C15⋊7Q16, D12⋊5D5, C3×Q8⋊2D5, D20.14D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C4○D8, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, Q8.13D6, C2×S3×D5, SD16⋊3D5, D5×C3⋊D4, D20.14D6
►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 23)(2 22)(3 21)(4 40)(5 39)(6 38)(7 37)(8 36)(9 35)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 27)(18 26)(19 25)(20 24)(41 131)(42 130)(43 129)(44 128)(45 127)(46 126)(47 125)(48 124)(49 123)(50 122)(51 121)(52 140)(53 139)(54 138)(55 137)(56 136)(57 135)(58 134)(59 133)(60 132)(61 219)(62 218)(63 217)(64 216)(65 215)(66 214)(67 213)(68 212)(69 211)(70 210)(71 209)(72 208)(73 207)(74 206)(75 205)(76 204)(77 203)(78 202)(79 201)(80 220)(81 102)(82 101)(83 120)(84 119)(85 118)(86 117)(87 116)(88 115)(89 114)(90 113)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(141 161)(142 180)(143 179)(144 178)(145 177)(146 176)(147 175)(148 174)(149 173)(150 172)(151 171)(152 170)(153 169)(154 168)(155 167)(156 166)(157 165)(158 164)(159 163)(160 162)(181 223)(182 222)(183 221)(184 240)(185 239)(186 238)(187 237)(188 236)(189 235)(190 234)(191 233)(192 232)(193 231)(194 230)(195 229)(196 228)(197 227)(198 226)(199 225)(200 224)
(1 43 186)(2 52 187 10 44 195)(3 41 188 19 45 184)(4 50 189 8 46 193)(5 59 190 17 47 182)(6 48 191)(7 57 192 15 49 200)(9 55 194 13 51 198)(11 53 196)(12 42 197 20 54 185)(14 60 199 18 56 183)(16 58 181)(21 133 236 27 127 222)(22 122 237 36 128 231)(23 131 238 25 129 240)(24 140 239 34 130 229)(26 138 221 32 132 227)(28 136 223 30 134 225)(29 125 224 39 135 234)(31 123 226 37 137 232)(33 121 228 35 139 230)(38 126 233 40 124 235)(61 172 112 69 164 120)(62 161 113 78 165 109)(63 170 114 67 166 118)(64 179 115 76 167 107)(65 168 116)(66 177 117 74 169 105)(68 175 119 72 171 103)(70 173 101)(71 162 102 79 174 110)(73 180 104 77 176 108)(75 178 106)(80 163 111)(81 203 148 95 209 142)(82 212 149 84 210 151)(83 201 150 93 211 160)(85 219 152 91 213 158)(86 208 153 100 214 147)(87 217 154 89 215 156)(88 206 155 98 216 145)(90 204 157 96 218 143)(92 202 159 94 220 141)(97 207 144 99 205 146)
(1 206 11 216)(2 205 12 215)(3 204 13 214)(4 203 14 213)(5 202 15 212)(6 201 16 211)(7 220 17 210)(8 219 18 209)(9 218 19 208)(10 217 20 207)(21 63 31 73)(22 62 32 72)(23 61 33 71)(24 80 34 70)(25 79 35 69)(26 78 36 68)(27 77 37 67)(28 76 38 66)(29 75 39 65)(30 74 40 64)(41 100 51 90)(42 99 52 89)(43 98 53 88)(44 97 54 87)(45 96 55 86)(46 95 56 85)(47 94 57 84)(48 93 58 83)(49 92 59 82)(50 91 60 81)(101 130 111 140)(102 129 112 139)(103 128 113 138)(104 127 114 137)(105 126 115 136)(106 125 116 135)(107 124 117 134)(108 123 118 133)(109 122 119 132)(110 121 120 131)(141 200 151 190)(142 199 152 189)(143 198 153 188)(144 197 154 187)(145 196 155 186)(146 195 156 185)(147 194 157 184)(148 193 158 183)(149 192 159 182)(150 191 160 181)(161 231 171 221)(162 230 172 240)(163 229 173 239)(164 228 174 238)(165 227 175 237)(166 226 176 236)(167 225 177 235)(168 224 178 234)(169 223 179 233)(170 222 180 232)
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), (1,23)(2,22)(3,21)(4,40)(5,39)(6,38)(7,37)(8,36)(9,35)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(41,131)(42,130)(43,129)(44,128)(45,127)(46,126)(47,125)(48,124)(49,123)(50,122)(51,121)(52,140)(53,139)(54,138)(55,137)(56,136)(57,135)(58,134)(59,133)(60,132)(61,219)(62,218)(63,217)(64,216)(65,215)(66,214)(67,213)(68,212)(69,211)(70,210)(71,209)(72,208)(73,207)(74,206)(75,205)(76,204)(77,203)(78,202)(79,201)(80,220)(81,102)(82,101)(83,120)(84,119)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(141,161)(142,180)(143,179)(144,178)(145,177)(146,176)(147,175)(148,174)(149,173)(150,172)(151,171)(152,170)(153,169)(154,168)(155,167)(156,166)(157,165)(158,164)(159,163)(160,162)(181,223)(182,222)(183,221)(184,240)(185,239)(186,238)(187,237)(188,236)(189,235)(190,234)(191,233)(192,232)(193,231)(194,230)(195,229)(196,228)(197,227)(198,226)(199,225)(200,224), (1,43,186)(2,52,187,10,44,195)(3,41,188,19,45,184)(4,50,189,8,46,193)(5,59,190,17,47,182)(6,48,191)(7,57,192,15,49,200)(9,55,194,13,51,198)(11,53,196)(12,42,197,20,54,185)(14,60,199,18,56,183)(16,58,181)(21,133,236,27,127,222)(22,122,237,36,128,231)(23,131,238,25,129,240)(24,140,239,34,130,229)(26,138,221,32,132,227)(28,136,223,30,134,225)(29,125,224,39,135,234)(31,123,226,37,137,232)(33,121,228,35,139,230)(38,126,233,40,124,235)(61,172,112,69,164,120)(62,161,113,78,165,109)(63,170,114,67,166,118)(64,179,115,76,167,107)(65,168,116)(66,177,117,74,169,105)(68,175,119,72,171,103)(70,173,101)(71,162,102,79,174,110)(73,180,104,77,176,108)(75,178,106)(80,163,111)(81,203,148,95,209,142)(82,212,149,84,210,151)(83,201,150,93,211,160)(85,219,152,91,213,158)(86,208,153,100,214,147)(87,217,154,89,215,156)(88,206,155,98,216,145)(90,204,157,96,218,143)(92,202,159,94,220,141)(97,207,144,99,205,146), (1,206,11,216)(2,205,12,215)(3,204,13,214)(4,203,14,213)(5,202,15,212)(6,201,16,211)(7,220,17,210)(8,219,18,209)(9,218,19,208)(10,217,20,207)(21,63,31,73)(22,62,32,72)(23,61,33,71)(24,80,34,70)(25,79,35,69)(26,78,36,68)(27,77,37,67)(28,76,38,66)(29,75,39,65)(30,74,40,64)(41,100,51,90)(42,99,52,89)(43,98,53,88)(44,97,54,87)(45,96,55,86)(46,95,56,85)(47,94,57,84)(48,93,58,83)(49,92,59,82)(50,91,60,81)(101,130,111,140)(102,129,112,139)(103,128,113,138)(104,127,114,137)(105,126,115,136)(106,125,116,135)(107,124,117,134)(108,123,118,133)(109,122,119,132)(110,121,120,131)(141,200,151,190)(142,199,152,189)(143,198,153,188)(144,197,154,187)(145,196,155,186)(146,195,156,185)(147,194,157,184)(148,193,158,183)(149,192,159,182)(150,191,160,181)(161,231,171,221)(162,230,172,240)(163,229,173,239)(164,228,174,238)(165,227,175,237)(166,226,176,236)(167,225,177,235)(168,224,178,234)(169,223,179,233)(170,222,180,232)>;
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), (1,23)(2,22)(3,21)(4,40)(5,39)(6,38)(7,37)(8,36)(9,35)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(41,131)(42,130)(43,129)(44,128)(45,127)(46,126)(47,125)(48,124)(49,123)(50,122)(51,121)(52,140)(53,139)(54,138)(55,137)(56,136)(57,135)(58,134)(59,133)(60,132)(61,219)(62,218)(63,217)(64,216)(65,215)(66,214)(67,213)(68,212)(69,211)(70,210)(71,209)(72,208)(73,207)(74,206)(75,205)(76,204)(77,203)(78,202)(79,201)(80,220)(81,102)(82,101)(83,120)(84,119)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(141,161)(142,180)(143,179)(144,178)(145,177)(146,176)(147,175)(148,174)(149,173)(150,172)(151,171)(152,170)(153,169)(154,168)(155,167)(156,166)(157,165)(158,164)(159,163)(160,162)(181,223)(182,222)(183,221)(184,240)(185,239)(186,238)(187,237)(188,236)(189,235)(190,234)(191,233)(192,232)(193,231)(194,230)(195,229)(196,228)(197,227)(198,226)(199,225)(200,224), (1,43,186)(2,52,187,10,44,195)(3,41,188,19,45,184)(4,50,189,8,46,193)(5,59,190,17,47,182)(6,48,191)(7,57,192,15,49,200)(9,55,194,13,51,198)(11,53,196)(12,42,197,20,54,185)(14,60,199,18,56,183)(16,58,181)(21,133,236,27,127,222)(22,122,237,36,128,231)(23,131,238,25,129,240)(24,140,239,34,130,229)(26,138,221,32,132,227)(28,136,223,30,134,225)(29,125,224,39,135,234)(31,123,226,37,137,232)(33,121,228,35,139,230)(38,126,233,40,124,235)(61,172,112,69,164,120)(62,161,113,78,165,109)(63,170,114,67,166,118)(64,179,115,76,167,107)(65,168,116)(66,177,117,74,169,105)(68,175,119,72,171,103)(70,173,101)(71,162,102,79,174,110)(73,180,104,77,176,108)(75,178,106)(80,163,111)(81,203,148,95,209,142)(82,212,149,84,210,151)(83,201,150,93,211,160)(85,219,152,91,213,158)(86,208,153,100,214,147)(87,217,154,89,215,156)(88,206,155,98,216,145)(90,204,157,96,218,143)(92,202,159,94,220,141)(97,207,144,99,205,146), (1,206,11,216)(2,205,12,215)(3,204,13,214)(4,203,14,213)(5,202,15,212)(6,201,16,211)(7,220,17,210)(8,219,18,209)(9,218,19,208)(10,217,20,207)(21,63,31,73)(22,62,32,72)(23,61,33,71)(24,80,34,70)(25,79,35,69)(26,78,36,68)(27,77,37,67)(28,76,38,66)(29,75,39,65)(30,74,40,64)(41,100,51,90)(42,99,52,89)(43,98,53,88)(44,97,54,87)(45,96,55,86)(46,95,56,85)(47,94,57,84)(48,93,58,83)(49,92,59,82)(50,91,60,81)(101,130,111,140)(102,129,112,139)(103,128,113,138)(104,127,114,137)(105,126,115,136)(106,125,116,135)(107,124,117,134)(108,123,118,133)(109,122,119,132)(110,121,120,131)(141,200,151,190)(142,199,152,189)(143,198,153,188)(144,197,154,187)(145,196,155,186)(146,195,156,185)(147,194,157,184)(148,193,158,183)(149,192,159,182)(150,191,160,181)(161,231,171,221)(162,230,172,240)(163,229,173,239)(164,228,174,238)(165,227,175,237)(166,226,176,236)(167,225,177,235)(168,224,178,234)(169,223,179,233)(170,222,180,232) );
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)], [(1,23),(2,22),(3,21),(4,40),(5,39),(6,38),(7,37),(8,36),(9,35),(10,34),(11,33),(12,32),(13,31),(14,30),(15,29),(16,28),(17,27),(18,26),(19,25),(20,24),(41,131),(42,130),(43,129),(44,128),(45,127),(46,126),(47,125),(48,124),(49,123),(50,122),(51,121),(52,140),(53,139),(54,138),(55,137),(56,136),(57,135),(58,134),(59,133),(60,132),(61,219),(62,218),(63,217),(64,216),(65,215),(66,214),(67,213),(68,212),(69,211),(70,210),(71,209),(72,208),(73,207),(74,206),(75,205),(76,204),(77,203),(78,202),(79,201),(80,220),(81,102),(82,101),(83,120),(84,119),(85,118),(86,117),(87,116),(88,115),(89,114),(90,113),(91,112),(92,111),(93,110),(94,109),(95,108),(96,107),(97,106),(98,105),(99,104),(100,103),(141,161),(142,180),(143,179),(144,178),(145,177),(146,176),(147,175),(148,174),(149,173),(150,172),(151,171),(152,170),(153,169),(154,168),(155,167),(156,166),(157,165),(158,164),(159,163),(160,162),(181,223),(182,222),(183,221),(184,240),(185,239),(186,238),(187,237),(188,236),(189,235),(190,234),(191,233),(192,232),(193,231),(194,230),(195,229),(196,228),(197,227),(198,226),(199,225),(200,224)], [(1,43,186),(2,52,187,10,44,195),(3,41,188,19,45,184),(4,50,189,8,46,193),(5,59,190,17,47,182),(6,48,191),(7,57,192,15,49,200),(9,55,194,13,51,198),(11,53,196),(12,42,197,20,54,185),(14,60,199,18,56,183),(16,58,181),(21,133,236,27,127,222),(22,122,237,36,128,231),(23,131,238,25,129,240),(24,140,239,34,130,229),(26,138,221,32,132,227),(28,136,223,30,134,225),(29,125,224,39,135,234),(31,123,226,37,137,232),(33,121,228,35,139,230),(38,126,233,40,124,235),(61,172,112,69,164,120),(62,161,113,78,165,109),(63,170,114,67,166,118),(64,179,115,76,167,107),(65,168,116),(66,177,117,74,169,105),(68,175,119,72,171,103),(70,173,101),(71,162,102,79,174,110),(73,180,104,77,176,108),(75,178,106),(80,163,111),(81,203,148,95,209,142),(82,212,149,84,210,151),(83,201,150,93,211,160),(85,219,152,91,213,158),(86,208,153,100,214,147),(87,217,154,89,215,156),(88,206,155,98,216,145),(90,204,157,96,218,143),(92,202,159,94,220,141),(97,207,144,99,205,146)], [(1,206,11,216),(2,205,12,215),(3,204,13,214),(4,203,14,213),(5,202,15,212),(6,201,16,211),(7,220,17,210),(8,219,18,209),(9,218,19,208),(10,217,20,207),(21,63,31,73),(22,62,32,72),(23,61,33,71),(24,80,34,70),(25,79,35,69),(26,78,36,68),(27,77,37,67),(28,76,38,66),(29,75,39,65),(30,74,40,64),(41,100,51,90),(42,99,52,89),(43,98,53,88),(44,97,54,87),(45,96,55,86),(46,95,56,85),(47,94,57,84),(48,93,58,83),(49,92,59,82),(50,91,60,81),(101,130,111,140),(102,129,112,139),(103,128,113,138),(104,127,114,137),(105,126,115,136),(106,125,116,135),(107,124,117,134),(108,123,118,133),(109,122,119,132),(110,121,120,131),(141,200,151,190),(142,199,152,189),(143,198,153,188),(144,197,154,187),(145,196,155,186),(146,195,156,185),(147,194,157,184),(148,193,158,183),(149,192,159,182),(150,191,160,181),(161,231,171,221),(162,230,172,240),(163,229,173,239),(164,228,174,238),(165,227,175,237),(166,226,176,236),(167,225,177,235),(168,224,178,234),(169,223,179,233),(170,222,180,232)])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | 40A | 40B | 40C | 40D | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 10 | 12 | 20 | 2 | 2 | 4 | 5 | 5 | 60 | 2 | 2 | 2 | 20 | 20 | 20 | 6 | 6 | 30 | 30 | 2 | 2 | 24 | 24 | 4 | 4 | 4 | 10 | 10 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | 4 | 12 | 12 | 12 | 12 | 8 | ··· | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | C3⋊D4 | C3⋊D4 | C4○D8 | S3×D5 | D4×D5 | Q8.13D6 | C2×S3×D5 | SD16⋊3D5 | D5×C3⋊D4 | D20.14D6 |
| kernel | D20.14D6 | D5×C3⋊C8 | C15⋊D8 | C6.D20 | C5×Q8⋊2S3 | C15⋊7Q16 | D12⋊5D5 | C3×Q8⋊2D5 | Q8⋊2D5 | C3×Dic5 | C6×D5 | Q8⋊2S3 | C4×D5 | D20 | C5×Q8 | C3⋊C8 | D12 | C3×Q8 | Dic5 | D10 | C15 | 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 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D20.14D6 ►in GL6(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 190 | 1 | 0 | 0 |
| 0 | 0 | 50 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 43 | 177 |
| 70 | 140 | 0 | 0 | 0 | 0 |
| 101 | 171 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 56 | 192 |
| 0 | 0 | 0 | 0 | 123 | 185 |
| 240 | 240 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 52 | 0 | 0 |
| 0 | 0 | 191 | 190 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 140 | 240 |
| 162 | 11 | 0 | 0 | 0 | 0 |
| 90 | 79 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 52 | 0 | 0 |
| 0 | 0 | 191 | 190 | 0 | 0 |
| 0 | 0 | 0 | 0 | 207 | 90 |
| 0 | 0 | 0 | 0 | 113 | 34 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,190,50,0,0,0,0,1,240,0,0,0,0,0,0,64,43,0,0,0,0,0,177],[70,101,0,0,0,0,140,171,0,0,0,0,0,0,1,0,0,0,0,0,1,240,0,0,0,0,0,0,56,123,0,0,0,0,192,185],[240,1,0,0,0,0,240,0,0,0,0,0,0,0,51,191,0,0,0,0,52,190,0,0,0,0,0,0,1,140,0,0,0,0,0,240],[162,90,0,0,0,0,11,79,0,0,0,0,0,0,51,191,0,0,0,0,52,190,0,0,0,0,0,0,207,113,0,0,0,0,90,34] >;
D20.14D6 in GAP, Magma, Sage, TeX
D_{20}._{14}D_6 % in TeX
G:=Group("D20.14D6"); // GroupNames label
G:=SmallGroup(480,590);
// by ID
G=gap.SmallGroup(480,590);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,422,135,100,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^6=1,d^2=a^10,b*a*b=d*a*d^-1=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^18*b,d*b*d^-1=a^13*b,d*c*d^-1=a^10*c^-1>;
// generators/relations