metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.28D4, C23.9D30, C2.8(D4×D15), C6.99(D4×D5), (C2×C4).6D30, C22⋊C4⋊3D15, C60⋊5C4⋊10C2, (C2×C20).32D6, D30⋊3C4⋊5C2, (C2×C12).33D10, C10.101(S3×D4), C30.307(C2×D4), C5⋊5(C23.9D6), C6.96(C4○D20), C30.38D4⋊4C2, C30.4Q8⋊10C2, (C22×C6).58D10, (C22×C10).73D6, C30.169(C4○D4), C10.96(C4○D12), C2.8(D4⋊2D15), C6.92(D4⋊2D5), C3⋊5(D10.12D4), (C2×C30).280C23, (C2×C60).173C22, C10.92(D4⋊2S3), C15⋊25(C22.D4), (C22×C30).14C22, C2.10(D60⋊11C2), C22.42(C22×D15), (C22×D15).80C22, (C2×Dic15).158C22, (C2×C4×D15)⋊16C2, (C5×C22⋊C4)⋊5S3, (C3×C22⋊C4)⋊5D5, (C15×C22⋊C4)⋊7C2, (C2×C15⋊7D4).3C2, (C2×C6).276(C22×D5), (C2×C10).275(C22×S3), SmallGroup(480,848)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30.28D4
G = < a,b,c,d | a30=b2=c4=1, d2=a15, bab=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd-1=a15c-1 >
Subgroups: 932 in 156 conjugacy classes, 49 normal (47 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, D15, C30, C30, C22.D4, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C2×C3⋊D4, Dic15, C60, D30, D30, C2×C30, C2×C30, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C2×C5⋊D4, C23.9D6, C4×D15, C2×Dic15, C15⋊7D4, C2×C60, C22×D15, C22×C30, D10.12D4, C30.4Q8, C60⋊5C4, D30⋊3C4, C30.38D4, C15×C22⋊C4, C2×C4×D15, C2×C15⋊7D4, D30.28D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C22×S3, D15, C22.D4, C22×D5, C4○D12, S3×D4, D4⋊2S3, D30, C4○D20, D4×D5, D4⋊2D5, C23.9D6, C22×D15, D10.12D4, D60⋊11C2, D4×D15, D4⋊2D15, D30.28D4
►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 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(50 60)(51 59)(52 58)(53 57)(54 56)(61 65)(62 64)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(101 102)(113 120)(114 119)(115 118)(116 117)(121 134)(122 133)(123 132)(124 131)(125 130)(126 129)(127 128)(135 150)(136 149)(137 148)(138 147)(139 146)(140 145)(141 144)(142 143)(151 171)(152 170)(153 169)(154 168)(155 167)(156 166)(157 165)(158 164)(159 163)(160 162)(172 180)(173 179)(174 178)(175 177)(181 201)(182 200)(183 199)(184 198)(185 197)(186 196)(187 195)(188 194)(189 193)(190 192)(202 210)(203 209)(204 208)(205 207)(211 226)(212 225)(213 224)(214 223)(215 222)(216 221)(217 220)(218 219)(227 240)(228 239)(229 238)(230 237)(231 236)(232 235)(233 234)
(1 169 102 86)(2 170 103 87)(3 171 104 88)(4 172 105 89)(5 173 106 90)(6 174 107 61)(7 175 108 62)(8 176 109 63)(9 177 110 64)(10 178 111 65)(11 179 112 66)(12 180 113 67)(13 151 114 68)(14 152 115 69)(15 153 116 70)(16 154 117 71)(17 155 118 72)(18 156 119 73)(19 157 120 74)(20 158 91 75)(21 159 92 76)(22 160 93 77)(23 161 94 78)(24 162 95 79)(25 163 96 80)(26 164 97 81)(27 165 98 82)(28 166 99 83)(29 167 100 84)(30 168 101 85)(31 126 197 217)(32 127 198 218)(33 128 199 219)(34 129 200 220)(35 130 201 221)(36 131 202 222)(37 132 203 223)(38 133 204 224)(39 134 205 225)(40 135 206 226)(41 136 207 227)(42 137 208 228)(43 138 209 229)(44 139 210 230)(45 140 181 231)(46 141 182 232)(47 142 183 233)(48 143 184 234)(49 144 185 235)(50 145 186 236)(51 146 187 237)(52 147 188 238)(53 148 189 239)(54 149 190 240)(55 150 191 211)(56 121 192 212)(57 122 193 213)(58 123 194 214)(59 124 195 215)(60 125 196 216)
(1 219 16 234)(2 220 17 235)(3 221 18 236)(4 222 19 237)(5 223 20 238)(6 224 21 239)(7 225 22 240)(8 226 23 211)(9 227 24 212)(10 228 25 213)(11 229 26 214)(12 230 27 215)(13 231 28 216)(14 232 29 217)(15 233 30 218)(31 84 46 69)(32 85 47 70)(33 86 48 71)(34 87 49 72)(35 88 50 73)(36 89 51 74)(37 90 52 75)(38 61 53 76)(39 62 54 77)(40 63 55 78)(41 64 56 79)(42 65 57 80)(43 66 58 81)(44 67 59 82)(45 68 60 83)(91 147 106 132)(92 148 107 133)(93 149 108 134)(94 150 109 135)(95 121 110 136)(96 122 111 137)(97 123 112 138)(98 124 113 139)(99 125 114 140)(100 126 115 141)(101 127 116 142)(102 128 117 143)(103 129 118 144)(104 130 119 145)(105 131 120 146)(151 196 166 181)(152 197 167 182)(153 198 168 183)(154 199 169 184)(155 200 170 185)(156 201 171 186)(157 202 172 187)(158 203 173 188)(159 204 174 189)(160 205 175 190)(161 206 176 191)(162 207 177 192)(163 208 178 193)(164 209 179 194)(165 210 180 195)
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)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(50,60)(51,59)(52,58)(53,57)(54,56)(61,65)(62,64)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)(113,120)(114,119)(115,118)(116,117)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(135,150)(136,149)(137,148)(138,147)(139,146)(140,145)(141,144)(142,143)(151,171)(152,170)(153,169)(154,168)(155,167)(156,166)(157,165)(158,164)(159,163)(160,162)(172,180)(173,179)(174,178)(175,177)(181,201)(182,200)(183,199)(184,198)(185,197)(186,196)(187,195)(188,194)(189,193)(190,192)(202,210)(203,209)(204,208)(205,207)(211,226)(212,225)(213,224)(214,223)(215,222)(216,221)(217,220)(218,219)(227,240)(228,239)(229,238)(230,237)(231,236)(232,235)(233,234), (1,169,102,86)(2,170,103,87)(3,171,104,88)(4,172,105,89)(5,173,106,90)(6,174,107,61)(7,175,108,62)(8,176,109,63)(9,177,110,64)(10,178,111,65)(11,179,112,66)(12,180,113,67)(13,151,114,68)(14,152,115,69)(15,153,116,70)(16,154,117,71)(17,155,118,72)(18,156,119,73)(19,157,120,74)(20,158,91,75)(21,159,92,76)(22,160,93,77)(23,161,94,78)(24,162,95,79)(25,163,96,80)(26,164,97,81)(27,165,98,82)(28,166,99,83)(29,167,100,84)(30,168,101,85)(31,126,197,217)(32,127,198,218)(33,128,199,219)(34,129,200,220)(35,130,201,221)(36,131,202,222)(37,132,203,223)(38,133,204,224)(39,134,205,225)(40,135,206,226)(41,136,207,227)(42,137,208,228)(43,138,209,229)(44,139,210,230)(45,140,181,231)(46,141,182,232)(47,142,183,233)(48,143,184,234)(49,144,185,235)(50,145,186,236)(51,146,187,237)(52,147,188,238)(53,148,189,239)(54,149,190,240)(55,150,191,211)(56,121,192,212)(57,122,193,213)(58,123,194,214)(59,124,195,215)(60,125,196,216), (1,219,16,234)(2,220,17,235)(3,221,18,236)(4,222,19,237)(5,223,20,238)(6,224,21,239)(7,225,22,240)(8,226,23,211)(9,227,24,212)(10,228,25,213)(11,229,26,214)(12,230,27,215)(13,231,28,216)(14,232,29,217)(15,233,30,218)(31,84,46,69)(32,85,47,70)(33,86,48,71)(34,87,49,72)(35,88,50,73)(36,89,51,74)(37,90,52,75)(38,61,53,76)(39,62,54,77)(40,63,55,78)(41,64,56,79)(42,65,57,80)(43,66,58,81)(44,67,59,82)(45,68,60,83)(91,147,106,132)(92,148,107,133)(93,149,108,134)(94,150,109,135)(95,121,110,136)(96,122,111,137)(97,123,112,138)(98,124,113,139)(99,125,114,140)(100,126,115,141)(101,127,116,142)(102,128,117,143)(103,129,118,144)(104,130,119,145)(105,131,120,146)(151,196,166,181)(152,197,167,182)(153,198,168,183)(154,199,169,184)(155,200,170,185)(156,201,171,186)(157,202,172,187)(158,203,173,188)(159,204,174,189)(160,205,175,190)(161,206,176,191)(162,207,177,192)(163,208,178,193)(164,209,179,194)(165,210,180,195)>;
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)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(50,60)(51,59)(52,58)(53,57)(54,56)(61,65)(62,64)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)(113,120)(114,119)(115,118)(116,117)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(135,150)(136,149)(137,148)(138,147)(139,146)(140,145)(141,144)(142,143)(151,171)(152,170)(153,169)(154,168)(155,167)(156,166)(157,165)(158,164)(159,163)(160,162)(172,180)(173,179)(174,178)(175,177)(181,201)(182,200)(183,199)(184,198)(185,197)(186,196)(187,195)(188,194)(189,193)(190,192)(202,210)(203,209)(204,208)(205,207)(211,226)(212,225)(213,224)(214,223)(215,222)(216,221)(217,220)(218,219)(227,240)(228,239)(229,238)(230,237)(231,236)(232,235)(233,234), (1,169,102,86)(2,170,103,87)(3,171,104,88)(4,172,105,89)(5,173,106,90)(6,174,107,61)(7,175,108,62)(8,176,109,63)(9,177,110,64)(10,178,111,65)(11,179,112,66)(12,180,113,67)(13,151,114,68)(14,152,115,69)(15,153,116,70)(16,154,117,71)(17,155,118,72)(18,156,119,73)(19,157,120,74)(20,158,91,75)(21,159,92,76)(22,160,93,77)(23,161,94,78)(24,162,95,79)(25,163,96,80)(26,164,97,81)(27,165,98,82)(28,166,99,83)(29,167,100,84)(30,168,101,85)(31,126,197,217)(32,127,198,218)(33,128,199,219)(34,129,200,220)(35,130,201,221)(36,131,202,222)(37,132,203,223)(38,133,204,224)(39,134,205,225)(40,135,206,226)(41,136,207,227)(42,137,208,228)(43,138,209,229)(44,139,210,230)(45,140,181,231)(46,141,182,232)(47,142,183,233)(48,143,184,234)(49,144,185,235)(50,145,186,236)(51,146,187,237)(52,147,188,238)(53,148,189,239)(54,149,190,240)(55,150,191,211)(56,121,192,212)(57,122,193,213)(58,123,194,214)(59,124,195,215)(60,125,196,216), (1,219,16,234)(2,220,17,235)(3,221,18,236)(4,222,19,237)(5,223,20,238)(6,224,21,239)(7,225,22,240)(8,226,23,211)(9,227,24,212)(10,228,25,213)(11,229,26,214)(12,230,27,215)(13,231,28,216)(14,232,29,217)(15,233,30,218)(31,84,46,69)(32,85,47,70)(33,86,48,71)(34,87,49,72)(35,88,50,73)(36,89,51,74)(37,90,52,75)(38,61,53,76)(39,62,54,77)(40,63,55,78)(41,64,56,79)(42,65,57,80)(43,66,58,81)(44,67,59,82)(45,68,60,83)(91,147,106,132)(92,148,107,133)(93,149,108,134)(94,150,109,135)(95,121,110,136)(96,122,111,137)(97,123,112,138)(98,124,113,139)(99,125,114,140)(100,126,115,141)(101,127,116,142)(102,128,117,143)(103,129,118,144)(104,130,119,145)(105,131,120,146)(151,196,166,181)(152,197,167,182)(153,198,168,183)(154,199,169,184)(155,200,170,185)(156,201,171,186)(157,202,172,187)(158,203,173,188)(159,204,174,189)(160,205,175,190)(161,206,176,191)(162,207,177,192)(163,208,178,193)(164,209,179,194)(165,210,180,195) );
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),(31,49),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(50,60),(51,59),(52,58),(53,57),(54,56),(61,65),(62,64),(66,90),(67,89),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(91,112),(92,111),(93,110),(94,109),(95,108),(96,107),(97,106),(98,105),(99,104),(100,103),(101,102),(113,120),(114,119),(115,118),(116,117),(121,134),(122,133),(123,132),(124,131),(125,130),(126,129),(127,128),(135,150),(136,149),(137,148),(138,147),(139,146),(140,145),(141,144),(142,143),(151,171),(152,170),(153,169),(154,168),(155,167),(156,166),(157,165),(158,164),(159,163),(160,162),(172,180),(173,179),(174,178),(175,177),(181,201),(182,200),(183,199),(184,198),(185,197),(186,196),(187,195),(188,194),(189,193),(190,192),(202,210),(203,209),(204,208),(205,207),(211,226),(212,225),(213,224),(214,223),(215,222),(216,221),(217,220),(218,219),(227,240),(228,239),(229,238),(230,237),(231,236),(232,235),(233,234)], [(1,169,102,86),(2,170,103,87),(3,171,104,88),(4,172,105,89),(5,173,106,90),(6,174,107,61),(7,175,108,62),(8,176,109,63),(9,177,110,64),(10,178,111,65),(11,179,112,66),(12,180,113,67),(13,151,114,68),(14,152,115,69),(15,153,116,70),(16,154,117,71),(17,155,118,72),(18,156,119,73),(19,157,120,74),(20,158,91,75),(21,159,92,76),(22,160,93,77),(23,161,94,78),(24,162,95,79),(25,163,96,80),(26,164,97,81),(27,165,98,82),(28,166,99,83),(29,167,100,84),(30,168,101,85),(31,126,197,217),(32,127,198,218),(33,128,199,219),(34,129,200,220),(35,130,201,221),(36,131,202,222),(37,132,203,223),(38,133,204,224),(39,134,205,225),(40,135,206,226),(41,136,207,227),(42,137,208,228),(43,138,209,229),(44,139,210,230),(45,140,181,231),(46,141,182,232),(47,142,183,233),(48,143,184,234),(49,144,185,235),(50,145,186,236),(51,146,187,237),(52,147,188,238),(53,148,189,239),(54,149,190,240),(55,150,191,211),(56,121,192,212),(57,122,193,213),(58,123,194,214),(59,124,195,215),(60,125,196,216)], [(1,219,16,234),(2,220,17,235),(3,221,18,236),(4,222,19,237),(5,223,20,238),(6,224,21,239),(7,225,22,240),(8,226,23,211),(9,227,24,212),(10,228,25,213),(11,229,26,214),(12,230,27,215),(13,231,28,216),(14,232,29,217),(15,233,30,218),(31,84,46,69),(32,85,47,70),(33,86,48,71),(34,87,49,72),(35,88,50,73),(36,89,51,74),(37,90,52,75),(38,61,53,76),(39,62,54,77),(40,63,55,78),(41,64,56,79),(42,65,57,80),(43,66,58,81),(44,67,59,82),(45,68,60,83),(91,147,106,132),(92,148,107,133),(93,149,108,134),(94,150,109,135),(95,121,110,136),(96,122,111,137),(97,123,112,138),(98,124,113,139),(99,125,114,140),(100,126,115,141),(101,127,116,142),(102,128,117,143),(103,129,118,144),(104,130,119,145),(105,131,120,146),(151,196,166,181),(152,197,167,182),(153,198,168,183),(154,199,169,184),(155,200,170,185),(156,201,171,186),(157,202,172,187),(158,203,173,188),(159,204,174,189),(160,205,175,190),(161,206,176,191),(162,207,177,192),(163,208,178,193),(164,209,179,194),(165,210,180,195)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 30 | 30 | 2 | 2 | 2 | 4 | 30 | 30 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D15 | C4○D12 | D30 | D30 | C4○D20 | D60⋊11C2 | S3×D4 | D4⋊2S3 | D4×D5 | D4⋊2D5 | D4×D15 | D4⋊2D15 |
| kernel | D30.28D4 | C30.4Q8 | C60⋊5C4 | D30⋊3C4 | C30.38D4 | C15×C22⋊C4 | C2×C4×D15 | C2×C15⋊7D4 | C5×C22⋊C4 | D30 | C3×C22⋊C4 | C2×C20 | C22×C10 | C30 | C2×C12 | C22×C6 | C22⋊C4 | C10 | C2×C4 | C23 | C6 | C2 | C10 | C10 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 2 | 4 | 4 | 8 | 4 | 8 | 16 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D30.28D4 ►in GL8(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 53 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 49 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 53 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 50 | 39 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 0 | 0 | 50 | 0 |
| 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 11 |
G:=sub<GL(8,GF(61))| [60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,53,49,0,0,0,0,0,0,20,7,0,0,0,0,0,0,0,0,17,1,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[60,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,53,6,0,0,0,0,0,0,20,8,0,0,0,0,0,0,0,0,47,30,0,0,0,0,0,0,24,14,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1],[50,0,0,0,0,0,0,0,39,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,11,0],[11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,11] >;
D30.28D4 in GAP, Magma, Sage, TeX
D_{30}._{28}D_4 % in TeX
G:=Group("D30.28D4"); // GroupNames label
G:=SmallGroup(480,848);
// by ID
G=gap.SmallGroup(480,848);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,254,219,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=a^15,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=a^15*c^-1>;
// generators/relations