metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic15⋊8D4, D12⋊5Dic5, C20⋊7(C4×S3), C15⋊14(C4×D4), C3⋊2(D4×Dic5), C60⋊21(C2×C4), C4⋊2(S3×Dic5), C6.41(D4×D5), (C5×D12)⋊12C4, C4⋊Dic5⋊13S3, D6⋊2(C2×Dic5), (C2×D12).9D5, C12⋊2(C2×Dic5), C10.42(S3×D4), C30.53(C2×D4), C5⋊6(Dic3⋊5D4), D6⋊Dic5⋊11C2, (C10×D12).9C2, (C2×C20).127D6, (C4×Dic15)⋊26C2, C30.74(C4○D4), (C2×C12).128D10, C2.2(C20⋊D6), C6.14(D4⋊2D5), C2.5(D12⋊D5), (C2×C60).201C22, (C2×C30).125C23, C30.131(C22×C4), (C2×Dic5).113D6, (C22×S3).43D10, C6.13(C22×Dic5), C10.36(Q8⋊3S3), (C6×Dic5).77C22, (C2×Dic15).207C22, (C2×S3×Dic5)⋊7C2, C10.120(S3×C2×C4), (S3×C10)⋊11(C2×C4), C2.14(C2×S3×Dic5), C22.59(C2×S3×D5), (C3×C4⋊Dic5)⋊10C2, (C2×C4).211(S3×D5), (S3×C2×C10).24C22, (C2×C6).137(C22×D5), (C2×C10).137(C22×S3), SmallGroup(480,511)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic15⋊8D4
G = < a,b,c,d | a12=b2=c10=1, d2=c5, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=c-1 >
Subgroups: 812 in 188 conjugacy classes, 70 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, D4, C23, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C5×S3, C30, C4×D4, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, C4×Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C3×Dic5, Dic15, Dic15, C60, S3×C10, S3×C10, C2×C30, C4×Dic5, C4⋊Dic5, C23.D5, C22×Dic5, D4×C10, Dic3⋊5D4, S3×Dic5, C6×Dic5, C5×D12, C2×Dic15, C2×C60, S3×C2×C10, D4×Dic5, D6⋊Dic5, C3×C4⋊Dic5, C4×Dic15, C2×S3×Dic5, C10×D12, Dic15⋊8D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22×C4, C2×D4, C4○D4, Dic5, D10, C4×S3, C22×S3, C4×D4, C2×Dic5, C22×D5, S3×C2×C4, S3×D4, Q8⋊3S3, S3×D5, D4×D5, D4⋊2D5, C22×Dic5, Dic3⋊5D4, S3×Dic5, C2×S3×D5, D4×Dic5, D12⋊D5, C20⋊D6, C2×S3×Dic5, Dic15⋊8D4
►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 3)(4 12)(5 11)(6 10)(7 9)(13 23)(14 22)(15 21)(16 20)(17 19)(25 27)(28 36)(29 35)(30 34)(31 33)(37 45)(38 44)(39 43)(40 42)(46 48)(49 53)(50 52)(54 60)(55 59)(56 58)(61 71)(62 70)(63 69)(64 68)(65 67)(73 75)(76 84)(77 83)(78 82)(79 81)(85 87)(88 96)(89 95)(90 94)(91 93)(97 103)(98 102)(99 101)(104 108)(105 107)(109 115)(110 114)(111 113)(116 120)(117 119)(121 127)(122 126)(123 125)(128 132)(129 131)(134 144)(135 143)(136 142)(137 141)(138 140)(145 151)(146 150)(147 149)(152 156)(153 155)(157 165)(158 164)(159 163)(160 162)(166 168)(169 171)(172 180)(173 179)(174 178)(175 177)(182 192)(183 191)(184 190)(185 189)(186 188)(193 199)(194 198)(195 197)(200 204)(201 203)(205 213)(206 212)(207 211)(208 210)(214 216)(217 227)(218 226)(219 225)(220 224)(221 223)(229 237)(230 236)(231 235)(232 234)(238 240)
(1 160 238 192 46 99 195 227 129 25)(2 161 239 181 47 100 196 228 130 26)(3 162 240 182 48 101 197 217 131 27)(4 163 229 183 37 102 198 218 132 28)(5 164 230 184 38 103 199 219 121 29)(6 165 231 185 39 104 200 220 122 30)(7 166 232 186 40 105 201 221 123 31)(8 167 233 187 41 106 202 222 124 32)(9 168 234 188 42 107 203 223 125 33)(10 157 235 189 43 108 204 224 126 34)(11 158 236 190 44 97 193 225 127 35)(12 159 237 191 45 98 194 226 128 36)(13 171 52 140 155 93 75 216 67 119)(14 172 53 141 156 94 76 205 68 120)(15 173 54 142 145 95 77 206 69 109)(16 174 55 143 146 96 78 207 70 110)(17 175 56 144 147 85 79 208 71 111)(18 176 57 133 148 86 80 209 72 112)(19 177 58 134 149 87 81 210 61 113)(20 178 59 135 150 88 82 211 62 114)(21 179 60 136 151 89 83 212 63 115)(22 180 49 137 152 90 84 213 64 116)(23 169 50 138 153 91 73 214 65 117)(24 170 51 139 154 92 74 215 66 118)
(1 156 99 120)(2 151 100 115)(3 146 101 110)(4 153 102 117)(5 148 103 112)(6 155 104 119)(7 150 105 114)(8 145 106 109)(9 152 107 116)(10 147 108 111)(11 154 97 118)(12 149 98 113)(13 30 93 39)(14 25 94 46)(15 32 95 41)(16 27 96 48)(17 34 85 43)(18 29 86 38)(19 36 87 45)(20 31 88 40)(21 26 89 47)(22 33 90 42)(23 28 91 37)(24 35 92 44)(49 223 213 234)(50 218 214 229)(51 225 215 236)(52 220 216 231)(53 227 205 238)(54 222 206 233)(55 217 207 240)(56 224 208 235)(57 219 209 230)(58 226 210 237)(59 221 211 232)(60 228 212 239)(61 159 134 194)(62 166 135 201)(63 161 136 196)(64 168 137 203)(65 163 138 198)(66 158 139 193)(67 165 140 200)(68 160 141 195)(69 167 142 202)(70 162 143 197)(71 157 144 204)(72 164 133 199)(73 183 169 132)(74 190 170 127)(75 185 171 122)(76 192 172 129)(77 187 173 124)(78 182 174 131)(79 189 175 126)(80 184 176 121)(81 191 177 128)(82 186 178 123)(83 181 179 130)(84 188 180 125)
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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,27)(28,36)(29,35)(30,34)(31,33)(37,45)(38,44)(39,43)(40,42)(46,48)(49,53)(50,52)(54,60)(55,59)(56,58)(61,71)(62,70)(63,69)(64,68)(65,67)(73,75)(76,84)(77,83)(78,82)(79,81)(85,87)(88,96)(89,95)(90,94)(91,93)(97,103)(98,102)(99,101)(104,108)(105,107)(109,115)(110,114)(111,113)(116,120)(117,119)(121,127)(122,126)(123,125)(128,132)(129,131)(134,144)(135,143)(136,142)(137,141)(138,140)(145,151)(146,150)(147,149)(152,156)(153,155)(157,165)(158,164)(159,163)(160,162)(166,168)(169,171)(172,180)(173,179)(174,178)(175,177)(182,192)(183,191)(184,190)(185,189)(186,188)(193,199)(194,198)(195,197)(200,204)(201,203)(205,213)(206,212)(207,211)(208,210)(214,216)(217,227)(218,226)(219,225)(220,224)(221,223)(229,237)(230,236)(231,235)(232,234)(238,240), (1,160,238,192,46,99,195,227,129,25)(2,161,239,181,47,100,196,228,130,26)(3,162,240,182,48,101,197,217,131,27)(4,163,229,183,37,102,198,218,132,28)(5,164,230,184,38,103,199,219,121,29)(6,165,231,185,39,104,200,220,122,30)(7,166,232,186,40,105,201,221,123,31)(8,167,233,187,41,106,202,222,124,32)(9,168,234,188,42,107,203,223,125,33)(10,157,235,189,43,108,204,224,126,34)(11,158,236,190,44,97,193,225,127,35)(12,159,237,191,45,98,194,226,128,36)(13,171,52,140,155,93,75,216,67,119)(14,172,53,141,156,94,76,205,68,120)(15,173,54,142,145,95,77,206,69,109)(16,174,55,143,146,96,78,207,70,110)(17,175,56,144,147,85,79,208,71,111)(18,176,57,133,148,86,80,209,72,112)(19,177,58,134,149,87,81,210,61,113)(20,178,59,135,150,88,82,211,62,114)(21,179,60,136,151,89,83,212,63,115)(22,180,49,137,152,90,84,213,64,116)(23,169,50,138,153,91,73,214,65,117)(24,170,51,139,154,92,74,215,66,118), (1,156,99,120)(2,151,100,115)(3,146,101,110)(4,153,102,117)(5,148,103,112)(6,155,104,119)(7,150,105,114)(8,145,106,109)(9,152,107,116)(10,147,108,111)(11,154,97,118)(12,149,98,113)(13,30,93,39)(14,25,94,46)(15,32,95,41)(16,27,96,48)(17,34,85,43)(18,29,86,38)(19,36,87,45)(20,31,88,40)(21,26,89,47)(22,33,90,42)(23,28,91,37)(24,35,92,44)(49,223,213,234)(50,218,214,229)(51,225,215,236)(52,220,216,231)(53,227,205,238)(54,222,206,233)(55,217,207,240)(56,224,208,235)(57,219,209,230)(58,226,210,237)(59,221,211,232)(60,228,212,239)(61,159,134,194)(62,166,135,201)(63,161,136,196)(64,168,137,203)(65,163,138,198)(66,158,139,193)(67,165,140,200)(68,160,141,195)(69,167,142,202)(70,162,143,197)(71,157,144,204)(72,164,133,199)(73,183,169,132)(74,190,170,127)(75,185,171,122)(76,192,172,129)(77,187,173,124)(78,182,174,131)(79,189,175,126)(80,184,176,121)(81,191,177,128)(82,186,178,123)(83,181,179,130)(84,188,180,125)>;
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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,27)(28,36)(29,35)(30,34)(31,33)(37,45)(38,44)(39,43)(40,42)(46,48)(49,53)(50,52)(54,60)(55,59)(56,58)(61,71)(62,70)(63,69)(64,68)(65,67)(73,75)(76,84)(77,83)(78,82)(79,81)(85,87)(88,96)(89,95)(90,94)(91,93)(97,103)(98,102)(99,101)(104,108)(105,107)(109,115)(110,114)(111,113)(116,120)(117,119)(121,127)(122,126)(123,125)(128,132)(129,131)(134,144)(135,143)(136,142)(137,141)(138,140)(145,151)(146,150)(147,149)(152,156)(153,155)(157,165)(158,164)(159,163)(160,162)(166,168)(169,171)(172,180)(173,179)(174,178)(175,177)(182,192)(183,191)(184,190)(185,189)(186,188)(193,199)(194,198)(195,197)(200,204)(201,203)(205,213)(206,212)(207,211)(208,210)(214,216)(217,227)(218,226)(219,225)(220,224)(221,223)(229,237)(230,236)(231,235)(232,234)(238,240), (1,160,238,192,46,99,195,227,129,25)(2,161,239,181,47,100,196,228,130,26)(3,162,240,182,48,101,197,217,131,27)(4,163,229,183,37,102,198,218,132,28)(5,164,230,184,38,103,199,219,121,29)(6,165,231,185,39,104,200,220,122,30)(7,166,232,186,40,105,201,221,123,31)(8,167,233,187,41,106,202,222,124,32)(9,168,234,188,42,107,203,223,125,33)(10,157,235,189,43,108,204,224,126,34)(11,158,236,190,44,97,193,225,127,35)(12,159,237,191,45,98,194,226,128,36)(13,171,52,140,155,93,75,216,67,119)(14,172,53,141,156,94,76,205,68,120)(15,173,54,142,145,95,77,206,69,109)(16,174,55,143,146,96,78,207,70,110)(17,175,56,144,147,85,79,208,71,111)(18,176,57,133,148,86,80,209,72,112)(19,177,58,134,149,87,81,210,61,113)(20,178,59,135,150,88,82,211,62,114)(21,179,60,136,151,89,83,212,63,115)(22,180,49,137,152,90,84,213,64,116)(23,169,50,138,153,91,73,214,65,117)(24,170,51,139,154,92,74,215,66,118), (1,156,99,120)(2,151,100,115)(3,146,101,110)(4,153,102,117)(5,148,103,112)(6,155,104,119)(7,150,105,114)(8,145,106,109)(9,152,107,116)(10,147,108,111)(11,154,97,118)(12,149,98,113)(13,30,93,39)(14,25,94,46)(15,32,95,41)(16,27,96,48)(17,34,85,43)(18,29,86,38)(19,36,87,45)(20,31,88,40)(21,26,89,47)(22,33,90,42)(23,28,91,37)(24,35,92,44)(49,223,213,234)(50,218,214,229)(51,225,215,236)(52,220,216,231)(53,227,205,238)(54,222,206,233)(55,217,207,240)(56,224,208,235)(57,219,209,230)(58,226,210,237)(59,221,211,232)(60,228,212,239)(61,159,134,194)(62,166,135,201)(63,161,136,196)(64,168,137,203)(65,163,138,198)(66,158,139,193)(67,165,140,200)(68,160,141,195)(69,167,142,202)(70,162,143,197)(71,157,144,204)(72,164,133,199)(73,183,169,132)(74,190,170,127)(75,185,171,122)(76,192,172,129)(77,187,173,124)(78,182,174,131)(79,189,175,126)(80,184,176,121)(81,191,177,128)(82,186,178,123)(83,181,179,130)(84,188,180,125) );
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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,23),(14,22),(15,21),(16,20),(17,19),(25,27),(28,36),(29,35),(30,34),(31,33),(37,45),(38,44),(39,43),(40,42),(46,48),(49,53),(50,52),(54,60),(55,59),(56,58),(61,71),(62,70),(63,69),(64,68),(65,67),(73,75),(76,84),(77,83),(78,82),(79,81),(85,87),(88,96),(89,95),(90,94),(91,93),(97,103),(98,102),(99,101),(104,108),(105,107),(109,115),(110,114),(111,113),(116,120),(117,119),(121,127),(122,126),(123,125),(128,132),(129,131),(134,144),(135,143),(136,142),(137,141),(138,140),(145,151),(146,150),(147,149),(152,156),(153,155),(157,165),(158,164),(159,163),(160,162),(166,168),(169,171),(172,180),(173,179),(174,178),(175,177),(182,192),(183,191),(184,190),(185,189),(186,188),(193,199),(194,198),(195,197),(200,204),(201,203),(205,213),(206,212),(207,211),(208,210),(214,216),(217,227),(218,226),(219,225),(220,224),(221,223),(229,237),(230,236),(231,235),(232,234),(238,240)], [(1,160,238,192,46,99,195,227,129,25),(2,161,239,181,47,100,196,228,130,26),(3,162,240,182,48,101,197,217,131,27),(4,163,229,183,37,102,198,218,132,28),(5,164,230,184,38,103,199,219,121,29),(6,165,231,185,39,104,200,220,122,30),(7,166,232,186,40,105,201,221,123,31),(8,167,233,187,41,106,202,222,124,32),(9,168,234,188,42,107,203,223,125,33),(10,157,235,189,43,108,204,224,126,34),(11,158,236,190,44,97,193,225,127,35),(12,159,237,191,45,98,194,226,128,36),(13,171,52,140,155,93,75,216,67,119),(14,172,53,141,156,94,76,205,68,120),(15,173,54,142,145,95,77,206,69,109),(16,174,55,143,146,96,78,207,70,110),(17,175,56,144,147,85,79,208,71,111),(18,176,57,133,148,86,80,209,72,112),(19,177,58,134,149,87,81,210,61,113),(20,178,59,135,150,88,82,211,62,114),(21,179,60,136,151,89,83,212,63,115),(22,180,49,137,152,90,84,213,64,116),(23,169,50,138,153,91,73,214,65,117),(24,170,51,139,154,92,74,215,66,118)], [(1,156,99,120),(2,151,100,115),(3,146,101,110),(4,153,102,117),(5,148,103,112),(6,155,104,119),(7,150,105,114),(8,145,106,109),(9,152,107,116),(10,147,108,111),(11,154,97,118),(12,149,98,113),(13,30,93,39),(14,25,94,46),(15,32,95,41),(16,27,96,48),(17,34,85,43),(18,29,86,38),(19,36,87,45),(20,31,88,40),(21,26,89,47),(22,33,90,42),(23,28,91,37),(24,35,92,44),(49,223,213,234),(50,218,214,229),(51,225,215,236),(52,220,216,231),(53,227,205,238),(54,222,206,233),(55,217,207,240),(56,224,208,235),(57,219,209,230),(58,226,210,237),(59,221,211,232),(60,228,212,239),(61,159,134,194),(62,166,135,201),(63,161,136,196),(64,168,137,203),(65,163,138,198),(66,158,139,193),(67,165,140,200),(68,160,141,195),(69,167,142,202),(70,162,143,197),(71,157,144,204),(72,164,133,199),(73,183,169,132),(74,190,170,127),(75,185,171,122),(76,192,172,129),(77,187,173,124),(78,182,174,131),(79,189,175,126),(80,184,176,121),(81,191,177,128),(82,186,178,123),(83,181,179,130),(84,188,180,125)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 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 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | D6 | C4○D4 | Dic5 | D10 | D10 | C4×S3 | S3×D4 | Q8⋊3S3 | S3×D5 | D4×D5 | D4⋊2D5 | S3×Dic5 | C2×S3×D5 | D12⋊D5 | C20⋊D6 |
| kernel | Dic15⋊8D4 | D6⋊Dic5 | C3×C4⋊Dic5 | C4×Dic15 | C2×S3×Dic5 | C10×D12 | C5×D12 | C4⋊Dic5 | Dic15 | C2×D12 | C2×Dic5 | C2×C20 | C30 | D12 | C2×C12 | C22×S3 | C20 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 2 | 2 | 1 | 2 | 8 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of Dic15⋊8D4 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 18 | 60 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 30 | 53 | 0 | 0 | 0 | 0 |
| 44 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 60 | 0 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,60,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60],[18,1,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[30,44,0,0,0,0,53,31,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,0,60,0,0,0,0,60,0] >;
Dic15⋊8D4 in GAP, Magma, Sage, TeX
{\rm Dic}_{15}\rtimes_8D_4 % in TeX
G:=Group("Dic15:8D4"); // GroupNames label
G:=SmallGroup(480,511);
// by ID
G=gap.SmallGroup(480,511);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^10=1,d^2=c^5,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=c^-1>;
// generators/relations