metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.4D4, C40.26D6, Dic20⋊6S3, D12.5D10, C24.29D10, Dic6.5D10, Dic15.4D4, Dic10.5D6, C120.39C22, C60.148C23, C8.6(S3×D5), C24⋊C2⋊4D5, C6.37(D4×D5), C40⋊S3⋊6C2, D15⋊Q8⋊10C2, C10.37(S3×D4), C30.30(C2×D4), C5⋊2(Q16⋊S3), (C3×Dic20)⋊9C2, C15⋊Q16⋊12C2, C3⋊2(SD16⋊D5), C15⋊6(C8.C22), D12⋊D5.2C2, C20.D6⋊12C2, C20.83(C22×S3), C12.83(C22×D5), C2.15(C20⋊D6), C15⋊3C8.24C22, (C4×D15).33C22, (C5×D12).29C22, (C5×Dic6).31C22, (C3×Dic10).31C22, C4.121(C2×S3×D5), (C5×C24⋊C2)⋊6C2, SmallGroup(480,356)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30.4D4
G = < a,b,c,d | a30=b2=1, c4=d2=a15, bab=a-1, ac=ca, dad-1=a11, cbc-1=a15b, dbd-1=a10b, dcd-1=c3 >
Subgroups: 668 in 120 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C8, C8, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, C12, D6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, D12, C3×Q8, C5×S3, D15, C30, C8.C22, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C8⋊S3, C24⋊C2, Q8⋊2S3, C3⋊Q16, C3×Q16, S3×Q8, Q8⋊3S3, C5×Dic3, C3×Dic5, Dic15, C60, S3×C10, D30, C8⋊D5, Dic20, D4.D5, C5⋊Q16, C5×SD16, D4⋊2D5, Q8×D5, Q16⋊S3, C15⋊3C8, C120, S3×Dic5, D30.C2, C5⋊D12, C15⋊Q8, C3×Dic10, C5×Dic6, C5×D12, C4×D15, SD16⋊D5, C20.D6, C15⋊Q16, C3×Dic20, C5×C24⋊C2, C40⋊S3, D12⋊D5, D15⋊Q8, D30.4D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8.C22, C22×D5, S3×D4, S3×D5, D4×D5, Q16⋊S3, C2×S3×D5, SD16⋊D5, C20⋊D6, D30.4D4
►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 40)(32 39)(33 38)(34 37)(35 36)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 85)(62 84)(63 83)(64 82)(65 81)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(86 90)(87 89)(91 115)(92 114)(93 113)(94 112)(95 111)(96 110)(97 109)(98 108)(99 107)(100 106)(101 105)(102 104)(116 120)(117 119)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(136 150)(137 149)(138 148)(139 147)(140 146)(141 145)(142 144)(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)(181 190)(182 189)(183 188)(184 187)(185 186)(191 210)(192 209)(193 208)(194 207)(195 206)(196 205)(197 204)(198 203)(199 202)(200 201)(211 220)(212 219)(213 218)(214 217)(215 216)(221 240)(222 239)(223 238)(224 237)(225 236)(226 235)(227 234)(228 233)(229 232)(230 231)
(1 96 36 81 16 111 51 66)(2 97 37 82 17 112 52 67)(3 98 38 83 18 113 53 68)(4 99 39 84 19 114 54 69)(5 100 40 85 20 115 55 70)(6 101 41 86 21 116 56 71)(7 102 42 87 22 117 57 72)(8 103 43 88 23 118 58 73)(9 104 44 89 24 119 59 74)(10 105 45 90 25 120 60 75)(11 106 46 61 26 91 31 76)(12 107 47 62 27 92 32 77)(13 108 48 63 28 93 33 78)(14 109 49 64 29 94 34 79)(15 110 50 65 30 95 35 80)(121 186 166 216 136 201 151 231)(122 187 167 217 137 202 152 232)(123 188 168 218 138 203 153 233)(124 189 169 219 139 204 154 234)(125 190 170 220 140 205 155 235)(126 191 171 221 141 206 156 236)(127 192 172 222 142 207 157 237)(128 193 173 223 143 208 158 238)(129 194 174 224 144 209 159 239)(130 195 175 225 145 210 160 240)(131 196 176 226 146 181 161 211)(132 197 177 227 147 182 162 212)(133 198 178 228 148 183 163 213)(134 199 179 229 149 184 164 214)(135 200 180 230 150 185 165 215)
(1 231 16 216)(2 212 17 227)(3 223 18 238)(4 234 19 219)(5 215 20 230)(6 226 21 211)(7 237 22 222)(8 218 23 233)(9 229 24 214)(10 240 25 225)(11 221 26 236)(12 232 27 217)(13 213 28 228)(14 224 29 239)(15 235 30 220)(31 206 46 191)(32 187 47 202)(33 198 48 183)(34 209 49 194)(35 190 50 205)(36 201 51 186)(37 182 52 197)(38 193 53 208)(39 204 54 189)(40 185 55 200)(41 196 56 181)(42 207 57 192)(43 188 58 203)(44 199 59 184)(45 210 60 195)(61 141 76 126)(62 122 77 137)(63 133 78 148)(64 144 79 129)(65 125 80 140)(66 136 81 121)(67 147 82 132)(68 128 83 143)(69 139 84 124)(70 150 85 135)(71 131 86 146)(72 142 87 127)(73 123 88 138)(74 134 89 149)(75 145 90 130)(91 171 106 156)(92 152 107 167)(93 163 108 178)(94 174 109 159)(95 155 110 170)(96 166 111 151)(97 177 112 162)(98 158 113 173)(99 169 114 154)(100 180 115 165)(101 161 116 176)(102 172 117 157)(103 153 118 168)(104 164 119 179)(105 175 120 160)
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)(31,40)(32,39)(33,38)(34,37)(35,36)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,85)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(86,90)(87,89)(91,115)(92,114)(93,113)(94,112)(95,111)(96,110)(97,109)(98,108)(99,107)(100,106)(101,105)(102,104)(116,120)(117,119)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,150)(137,149)(138,148)(139,147)(140,146)(141,145)(142,144)(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)(181,190)(182,189)(183,188)(184,187)(185,186)(191,210)(192,209)(193,208)(194,207)(195,206)(196,205)(197,204)(198,203)(199,202)(200,201)(211,220)(212,219)(213,218)(214,217)(215,216)(221,240)(222,239)(223,238)(224,237)(225,236)(226,235)(227,234)(228,233)(229,232)(230,231), (1,96,36,81,16,111,51,66)(2,97,37,82,17,112,52,67)(3,98,38,83,18,113,53,68)(4,99,39,84,19,114,54,69)(5,100,40,85,20,115,55,70)(6,101,41,86,21,116,56,71)(7,102,42,87,22,117,57,72)(8,103,43,88,23,118,58,73)(9,104,44,89,24,119,59,74)(10,105,45,90,25,120,60,75)(11,106,46,61,26,91,31,76)(12,107,47,62,27,92,32,77)(13,108,48,63,28,93,33,78)(14,109,49,64,29,94,34,79)(15,110,50,65,30,95,35,80)(121,186,166,216,136,201,151,231)(122,187,167,217,137,202,152,232)(123,188,168,218,138,203,153,233)(124,189,169,219,139,204,154,234)(125,190,170,220,140,205,155,235)(126,191,171,221,141,206,156,236)(127,192,172,222,142,207,157,237)(128,193,173,223,143,208,158,238)(129,194,174,224,144,209,159,239)(130,195,175,225,145,210,160,240)(131,196,176,226,146,181,161,211)(132,197,177,227,147,182,162,212)(133,198,178,228,148,183,163,213)(134,199,179,229,149,184,164,214)(135,200,180,230,150,185,165,215), (1,231,16,216)(2,212,17,227)(3,223,18,238)(4,234,19,219)(5,215,20,230)(6,226,21,211)(7,237,22,222)(8,218,23,233)(9,229,24,214)(10,240,25,225)(11,221,26,236)(12,232,27,217)(13,213,28,228)(14,224,29,239)(15,235,30,220)(31,206,46,191)(32,187,47,202)(33,198,48,183)(34,209,49,194)(35,190,50,205)(36,201,51,186)(37,182,52,197)(38,193,53,208)(39,204,54,189)(40,185,55,200)(41,196,56,181)(42,207,57,192)(43,188,58,203)(44,199,59,184)(45,210,60,195)(61,141,76,126)(62,122,77,137)(63,133,78,148)(64,144,79,129)(65,125,80,140)(66,136,81,121)(67,147,82,132)(68,128,83,143)(69,139,84,124)(70,150,85,135)(71,131,86,146)(72,142,87,127)(73,123,88,138)(74,134,89,149)(75,145,90,130)(91,171,106,156)(92,152,107,167)(93,163,108,178)(94,174,109,159)(95,155,110,170)(96,166,111,151)(97,177,112,162)(98,158,113,173)(99,169,114,154)(100,180,115,165)(101,161,116,176)(102,172,117,157)(103,153,118,168)(104,164,119,179)(105,175,120,160)>;
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)(31,40)(32,39)(33,38)(34,37)(35,36)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,85)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(86,90)(87,89)(91,115)(92,114)(93,113)(94,112)(95,111)(96,110)(97,109)(98,108)(99,107)(100,106)(101,105)(102,104)(116,120)(117,119)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,150)(137,149)(138,148)(139,147)(140,146)(141,145)(142,144)(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)(181,190)(182,189)(183,188)(184,187)(185,186)(191,210)(192,209)(193,208)(194,207)(195,206)(196,205)(197,204)(198,203)(199,202)(200,201)(211,220)(212,219)(213,218)(214,217)(215,216)(221,240)(222,239)(223,238)(224,237)(225,236)(226,235)(227,234)(228,233)(229,232)(230,231), (1,96,36,81,16,111,51,66)(2,97,37,82,17,112,52,67)(3,98,38,83,18,113,53,68)(4,99,39,84,19,114,54,69)(5,100,40,85,20,115,55,70)(6,101,41,86,21,116,56,71)(7,102,42,87,22,117,57,72)(8,103,43,88,23,118,58,73)(9,104,44,89,24,119,59,74)(10,105,45,90,25,120,60,75)(11,106,46,61,26,91,31,76)(12,107,47,62,27,92,32,77)(13,108,48,63,28,93,33,78)(14,109,49,64,29,94,34,79)(15,110,50,65,30,95,35,80)(121,186,166,216,136,201,151,231)(122,187,167,217,137,202,152,232)(123,188,168,218,138,203,153,233)(124,189,169,219,139,204,154,234)(125,190,170,220,140,205,155,235)(126,191,171,221,141,206,156,236)(127,192,172,222,142,207,157,237)(128,193,173,223,143,208,158,238)(129,194,174,224,144,209,159,239)(130,195,175,225,145,210,160,240)(131,196,176,226,146,181,161,211)(132,197,177,227,147,182,162,212)(133,198,178,228,148,183,163,213)(134,199,179,229,149,184,164,214)(135,200,180,230,150,185,165,215), (1,231,16,216)(2,212,17,227)(3,223,18,238)(4,234,19,219)(5,215,20,230)(6,226,21,211)(7,237,22,222)(8,218,23,233)(9,229,24,214)(10,240,25,225)(11,221,26,236)(12,232,27,217)(13,213,28,228)(14,224,29,239)(15,235,30,220)(31,206,46,191)(32,187,47,202)(33,198,48,183)(34,209,49,194)(35,190,50,205)(36,201,51,186)(37,182,52,197)(38,193,53,208)(39,204,54,189)(40,185,55,200)(41,196,56,181)(42,207,57,192)(43,188,58,203)(44,199,59,184)(45,210,60,195)(61,141,76,126)(62,122,77,137)(63,133,78,148)(64,144,79,129)(65,125,80,140)(66,136,81,121)(67,147,82,132)(68,128,83,143)(69,139,84,124)(70,150,85,135)(71,131,86,146)(72,142,87,127)(73,123,88,138)(74,134,89,149)(75,145,90,130)(91,171,106,156)(92,152,107,167)(93,163,108,178)(94,174,109,159)(95,155,110,170)(96,166,111,151)(97,177,112,162)(98,158,113,173)(99,169,114,154)(100,180,115,165)(101,161,116,176)(102,172,117,157)(103,153,118,168)(104,164,119,179)(105,175,120,160) );
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),(31,40),(32,39),(33,38),(34,37),(35,36),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(61,85),(62,84),(63,83),(64,82),(65,81),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74),(86,90),(87,89),(91,115),(92,114),(93,113),(94,112),(95,111),(96,110),(97,109),(98,108),(99,107),(100,106),(101,105),(102,104),(116,120),(117,119),(121,135),(122,134),(123,133),(124,132),(125,131),(126,130),(127,129),(136,150),(137,149),(138,148),(139,147),(140,146),(141,145),(142,144),(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),(181,190),(182,189),(183,188),(184,187),(185,186),(191,210),(192,209),(193,208),(194,207),(195,206),(196,205),(197,204),(198,203),(199,202),(200,201),(211,220),(212,219),(213,218),(214,217),(215,216),(221,240),(222,239),(223,238),(224,237),(225,236),(226,235),(227,234),(228,233),(229,232),(230,231)], [(1,96,36,81,16,111,51,66),(2,97,37,82,17,112,52,67),(3,98,38,83,18,113,53,68),(4,99,39,84,19,114,54,69),(5,100,40,85,20,115,55,70),(6,101,41,86,21,116,56,71),(7,102,42,87,22,117,57,72),(8,103,43,88,23,118,58,73),(9,104,44,89,24,119,59,74),(10,105,45,90,25,120,60,75),(11,106,46,61,26,91,31,76),(12,107,47,62,27,92,32,77),(13,108,48,63,28,93,33,78),(14,109,49,64,29,94,34,79),(15,110,50,65,30,95,35,80),(121,186,166,216,136,201,151,231),(122,187,167,217,137,202,152,232),(123,188,168,218,138,203,153,233),(124,189,169,219,139,204,154,234),(125,190,170,220,140,205,155,235),(126,191,171,221,141,206,156,236),(127,192,172,222,142,207,157,237),(128,193,173,223,143,208,158,238),(129,194,174,224,144,209,159,239),(130,195,175,225,145,210,160,240),(131,196,176,226,146,181,161,211),(132,197,177,227,147,182,162,212),(133,198,178,228,148,183,163,213),(134,199,179,229,149,184,164,214),(135,200,180,230,150,185,165,215)], [(1,231,16,216),(2,212,17,227),(3,223,18,238),(4,234,19,219),(5,215,20,230),(6,226,21,211),(7,237,22,222),(8,218,23,233),(9,229,24,214),(10,240,25,225),(11,221,26,236),(12,232,27,217),(13,213,28,228),(14,224,29,239),(15,235,30,220),(31,206,46,191),(32,187,47,202),(33,198,48,183),(34,209,49,194),(35,190,50,205),(36,201,51,186),(37,182,52,197),(38,193,53,208),(39,204,54,189),(40,185,55,200),(41,196,56,181),(42,207,57,192),(43,188,58,203),(44,199,59,184),(45,210,60,195),(61,141,76,126),(62,122,77,137),(63,133,78,148),(64,144,79,129),(65,125,80,140),(66,136,81,121),(67,147,82,132),(68,128,83,143),(69,139,84,124),(70,150,85,135),(71,131,86,146),(72,142,87,127),(73,123,88,138),(74,134,89,149),(75,145,90,130),(91,171,106,156),(92,152,107,167),(93,163,108,178),(94,174,109,159),(95,155,110,170),(96,166,111,151),(97,177,112,162),(98,158,113,173),(99,169,114,154),(100,180,115,165),(101,161,116,176),(102,172,117,157),(103,153,118,168),(104,164,119,179),(105,175,120,160)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 24A | 24B | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 12 | 30 | 2 | 2 | 12 | 20 | 20 | 30 | 2 | 2 | 2 | 4 | 60 | 2 | 2 | 24 | 24 | 4 | 40 | 40 | 4 | 4 | 4 | 4 | 24 | 24 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | D10 | C8.C22 | S3×D4 | S3×D5 | D4×D5 | Q16⋊S3 | C2×S3×D5 | SD16⋊D5 | C20⋊D6 | D30.4D4 |
| kernel | D30.4D4 | C20.D6 | C15⋊Q16 | C3×Dic20 | C5×C24⋊C2 | C40⋊S3 | D12⋊D5 | D15⋊Q8 | Dic20 | Dic15 | D30 | C24⋊C2 | C40 | Dic10 | C24 | Dic6 | D12 | C15 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of D30.4D4 ►in GL4(𝔽241) generated by
| 51 | 240 | 190 | 1 |
| 1 | 0 | 240 | 0 |
| 51 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 190 | 1 | 0 | 0 |
| 51 | 51 | 0 | 0 |
| 190 | 1 | 51 | 240 |
| 51 | 51 | 190 | 190 |
| 49 | 33 | 147 | 0 |
| 208 | 45 | 0 | 147 |
| 94 | 0 | 196 | 33 |
| 0 | 94 | 208 | 192 |
| 231 | 76 | 25 | 203 |
| 165 | 10 | 38 | 15 |
| 15 | 38 | 10 | 165 |
| 203 | 25 | 76 | 231 |
G:=sub<GL(4,GF(241))| [51,1,51,1,240,0,240,0,190,240,0,0,1,0,0,0],[190,51,190,51,1,51,1,51,0,0,51,190,0,0,240,190],[49,208,94,0,33,45,0,94,147,0,196,208,0,147,33,192],[231,165,15,203,76,10,38,25,25,38,10,76,203,15,165,231] >;
D30.4D4 in GAP, Magma, Sage, TeX
D_{30}._4D_4 % in TeX
G:=Group("D30.4D4"); // GroupNames label
G:=SmallGroup(480,356);
// by ID
G=gap.SmallGroup(480,356);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,303,142,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=1,c^4=d^2=a^15,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,c*b*c^-1=a^15*b,d*b*d^-1=a^10*b,d*c*d^-1=c^3>;
// generators/relations