metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.11D4, D12.10D10, C60.23C23, Dic15.43D4, Dic10.10D6, Dic30.7C22, D4⋊S3⋊5D5, D4.D5⋊5S3, C3⋊C8.16D10, C6.73(D4×D5), C15⋊18(C4○D8), C3⋊3(D8⋊3D5), (C5×D4).11D6, C3⋊Dic20⋊4C2, D4.17(S3×D5), C10.74(S3×D4), C5⋊2C8.16D6, D4⋊2D15⋊3C2, D12⋊D5⋊1C2, D15⋊2C8⋊4C2, (C3×D4).11D10, D12.D5⋊4C2, C30.185(C2×D4), C5⋊3(Q8.7D6), C20.23(C22×S3), (C4×D15).7C22, (C5×D12).7C22, C12.23(C22×D5), (D4×C15).17C22, C2.26(D10⋊D6), (C3×Dic10).7C22, C4.23(C2×S3×D5), (C5×D4⋊S3)⋊7C2, (C3×D4.D5)⋊7C2, (C5×C3⋊C8).6C22, (C3×C5⋊2C8).6C22, SmallGroup(480,575)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30.11D4
G = < a,b,c,d | a30=b2=1, c4=d2=a15, bab=a-1, cac-1=dad-1=a19, cbc-1=a18b, dbd-1=a3b, dcd-1=c3 >
Subgroups: 700 in 124 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, D4, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C5×S3, D15, C30, C30, C4○D8, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C5×D4, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C3×SD16, D4⋊2S3, Q8⋊3S3, C3×Dic5, Dic15, Dic15, C60, S3×C10, D30, C2×C30, C8×D5, Dic20, D4.D5, D4.D5, C5×D8, D4⋊2D5, Q8.7D6, C5×C3⋊C8, C3×C5⋊2C8, S3×Dic5, C5⋊D12, C3×Dic10, C5×D12, Dic30, C4×D15, C2×Dic15, C15⋊7D4, D4×C15, D8⋊3D5, D15⋊2C8, D12.D5, C3⋊Dic20, C3×D4.D5, C5×D4⋊S3, D12⋊D5, D4⋊2D15, D30.11D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C4○D8, C22×D5, S3×D4, S3×D5, D4×D5, Q8.7D6, C2×S3×D5, D8⋊3D5, D10⋊D6, D30.11D4
►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 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(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)(121 145)(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)(181 210)(182 209)(183 208)(184 207)(185 206)(186 205)(187 204)(188 203)(189 202)(190 201)(191 200)(192 199)(193 198)(194 197)(195 196)(211 240)(212 239)(213 238)(214 237)(215 236)(216 235)(217 234)(218 233)(219 232)(220 231)(221 230)(222 229)(223 228)(224 227)(225 226)
(1 211 56 196 16 226 41 181)(2 230 57 185 17 215 42 200)(3 219 58 204 18 234 43 189)(4 238 59 193 19 223 44 208)(5 227 60 182 20 212 45 197)(6 216 31 201 21 231 46 186)(7 235 32 190 22 220 47 205)(8 224 33 209 23 239 48 194)(9 213 34 198 24 228 49 183)(10 232 35 187 25 217 50 202)(11 221 36 206 26 236 51 191)(12 240 37 195 27 225 52 210)(13 229 38 184 28 214 53 199)(14 218 39 203 29 233 54 188)(15 237 40 192 30 222 55 207)(61 156 111 141 76 171 96 126)(62 175 112 130 77 160 97 145)(63 164 113 149 78 179 98 134)(64 153 114 138 79 168 99 123)(65 172 115 127 80 157 100 142)(66 161 116 146 81 176 101 131)(67 180 117 135 82 165 102 150)(68 169 118 124 83 154 103 139)(69 158 119 143 84 173 104 128)(70 177 120 132 85 162 105 147)(71 166 91 121 86 151 106 136)(72 155 92 140 87 170 107 125)(73 174 93 129 88 159 108 144)(74 163 94 148 89 178 109 133)(75 152 95 137 90 167 110 122)
(1 141 16 126)(2 130 17 145)(3 149 18 134)(4 138 19 123)(5 127 20 142)(6 146 21 131)(7 135 22 150)(8 124 23 139)(9 143 24 128)(10 132 25 147)(11 121 26 136)(12 140 27 125)(13 129 28 144)(14 148 29 133)(15 137 30 122)(31 161 46 176)(32 180 47 165)(33 169 48 154)(34 158 49 173)(35 177 50 162)(36 166 51 151)(37 155 52 170)(38 174 53 159)(39 163 54 178)(40 152 55 167)(41 171 56 156)(42 160 57 175)(43 179 58 164)(44 168 59 153)(45 157 60 172)(61 196 76 181)(62 185 77 200)(63 204 78 189)(64 193 79 208)(65 182 80 197)(66 201 81 186)(67 190 82 205)(68 209 83 194)(69 198 84 183)(70 187 85 202)(71 206 86 191)(72 195 87 210)(73 184 88 199)(74 203 89 188)(75 192 90 207)(91 221 106 236)(92 240 107 225)(93 229 108 214)(94 218 109 233)(95 237 110 222)(96 226 111 211)(97 215 112 230)(98 234 113 219)(99 223 114 238)(100 212 115 227)(101 231 116 216)(102 220 117 235)(103 239 118 224)(104 228 119 213)(105 217 120 232)
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), (1,30)(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)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(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)(121,145)(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)(181,210)(182,209)(183,208)(184,207)(185,206)(186,205)(187,204)(188,203)(189,202)(190,201)(191,200)(192,199)(193,198)(194,197)(195,196)(211,240)(212,239)(213,238)(214,237)(215,236)(216,235)(217,234)(218,233)(219,232)(220,231)(221,230)(222,229)(223,228)(224,227)(225,226), (1,211,56,196,16,226,41,181)(2,230,57,185,17,215,42,200)(3,219,58,204,18,234,43,189)(4,238,59,193,19,223,44,208)(5,227,60,182,20,212,45,197)(6,216,31,201,21,231,46,186)(7,235,32,190,22,220,47,205)(8,224,33,209,23,239,48,194)(9,213,34,198,24,228,49,183)(10,232,35,187,25,217,50,202)(11,221,36,206,26,236,51,191)(12,240,37,195,27,225,52,210)(13,229,38,184,28,214,53,199)(14,218,39,203,29,233,54,188)(15,237,40,192,30,222,55,207)(61,156,111,141,76,171,96,126)(62,175,112,130,77,160,97,145)(63,164,113,149,78,179,98,134)(64,153,114,138,79,168,99,123)(65,172,115,127,80,157,100,142)(66,161,116,146,81,176,101,131)(67,180,117,135,82,165,102,150)(68,169,118,124,83,154,103,139)(69,158,119,143,84,173,104,128)(70,177,120,132,85,162,105,147)(71,166,91,121,86,151,106,136)(72,155,92,140,87,170,107,125)(73,174,93,129,88,159,108,144)(74,163,94,148,89,178,109,133)(75,152,95,137,90,167,110,122), (1,141,16,126)(2,130,17,145)(3,149,18,134)(4,138,19,123)(5,127,20,142)(6,146,21,131)(7,135,22,150)(8,124,23,139)(9,143,24,128)(10,132,25,147)(11,121,26,136)(12,140,27,125)(13,129,28,144)(14,148,29,133)(15,137,30,122)(31,161,46,176)(32,180,47,165)(33,169,48,154)(34,158,49,173)(35,177,50,162)(36,166,51,151)(37,155,52,170)(38,174,53,159)(39,163,54,178)(40,152,55,167)(41,171,56,156)(42,160,57,175)(43,179,58,164)(44,168,59,153)(45,157,60,172)(61,196,76,181)(62,185,77,200)(63,204,78,189)(64,193,79,208)(65,182,80,197)(66,201,81,186)(67,190,82,205)(68,209,83,194)(69,198,84,183)(70,187,85,202)(71,206,86,191)(72,195,87,210)(73,184,88,199)(74,203,89,188)(75,192,90,207)(91,221,106,236)(92,240,107,225)(93,229,108,214)(94,218,109,233)(95,237,110,222)(96,226,111,211)(97,215,112,230)(98,234,113,219)(99,223,114,238)(100,212,115,227)(101,231,116,216)(102,220,117,235)(103,239,118,224)(104,228,119,213)(105,217,120,232)>;
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), (1,30)(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)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(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)(121,145)(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)(181,210)(182,209)(183,208)(184,207)(185,206)(186,205)(187,204)(188,203)(189,202)(190,201)(191,200)(192,199)(193,198)(194,197)(195,196)(211,240)(212,239)(213,238)(214,237)(215,236)(216,235)(217,234)(218,233)(219,232)(220,231)(221,230)(222,229)(223,228)(224,227)(225,226), (1,211,56,196,16,226,41,181)(2,230,57,185,17,215,42,200)(3,219,58,204,18,234,43,189)(4,238,59,193,19,223,44,208)(5,227,60,182,20,212,45,197)(6,216,31,201,21,231,46,186)(7,235,32,190,22,220,47,205)(8,224,33,209,23,239,48,194)(9,213,34,198,24,228,49,183)(10,232,35,187,25,217,50,202)(11,221,36,206,26,236,51,191)(12,240,37,195,27,225,52,210)(13,229,38,184,28,214,53,199)(14,218,39,203,29,233,54,188)(15,237,40,192,30,222,55,207)(61,156,111,141,76,171,96,126)(62,175,112,130,77,160,97,145)(63,164,113,149,78,179,98,134)(64,153,114,138,79,168,99,123)(65,172,115,127,80,157,100,142)(66,161,116,146,81,176,101,131)(67,180,117,135,82,165,102,150)(68,169,118,124,83,154,103,139)(69,158,119,143,84,173,104,128)(70,177,120,132,85,162,105,147)(71,166,91,121,86,151,106,136)(72,155,92,140,87,170,107,125)(73,174,93,129,88,159,108,144)(74,163,94,148,89,178,109,133)(75,152,95,137,90,167,110,122), (1,141,16,126)(2,130,17,145)(3,149,18,134)(4,138,19,123)(5,127,20,142)(6,146,21,131)(7,135,22,150)(8,124,23,139)(9,143,24,128)(10,132,25,147)(11,121,26,136)(12,140,27,125)(13,129,28,144)(14,148,29,133)(15,137,30,122)(31,161,46,176)(32,180,47,165)(33,169,48,154)(34,158,49,173)(35,177,50,162)(36,166,51,151)(37,155,52,170)(38,174,53,159)(39,163,54,178)(40,152,55,167)(41,171,56,156)(42,160,57,175)(43,179,58,164)(44,168,59,153)(45,157,60,172)(61,196,76,181)(62,185,77,200)(63,204,78,189)(64,193,79,208)(65,182,80,197)(66,201,81,186)(67,190,82,205)(68,209,83,194)(69,198,84,183)(70,187,85,202)(71,206,86,191)(72,195,87,210)(73,184,88,199)(74,203,89,188)(75,192,90,207)(91,221,106,236)(92,240,107,225)(93,229,108,214)(94,218,109,233)(95,237,110,222)(96,226,111,211)(97,215,112,230)(98,234,113,219)(99,223,114,238)(100,212,115,227)(101,231,116,216)(102,220,117,235)(103,239,118,224)(104,228,119,213)(105,217,120,232) );
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)], [(1,30),(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),(61,75),(62,74),(63,73),(64,72),(65,71),(66,70),(67,69),(76,90),(77,89),(78,88),(79,87),(80,86),(81,85),(82,84),(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),(121,145),(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),(181,210),(182,209),(183,208),(184,207),(185,206),(186,205),(187,204),(188,203),(189,202),(190,201),(191,200),(192,199),(193,198),(194,197),(195,196),(211,240),(212,239),(213,238),(214,237),(215,236),(216,235),(217,234),(218,233),(219,232),(220,231),(221,230),(222,229),(223,228),(224,227),(225,226)], [(1,211,56,196,16,226,41,181),(2,230,57,185,17,215,42,200),(3,219,58,204,18,234,43,189),(4,238,59,193,19,223,44,208),(5,227,60,182,20,212,45,197),(6,216,31,201,21,231,46,186),(7,235,32,190,22,220,47,205),(8,224,33,209,23,239,48,194),(9,213,34,198,24,228,49,183),(10,232,35,187,25,217,50,202),(11,221,36,206,26,236,51,191),(12,240,37,195,27,225,52,210),(13,229,38,184,28,214,53,199),(14,218,39,203,29,233,54,188),(15,237,40,192,30,222,55,207),(61,156,111,141,76,171,96,126),(62,175,112,130,77,160,97,145),(63,164,113,149,78,179,98,134),(64,153,114,138,79,168,99,123),(65,172,115,127,80,157,100,142),(66,161,116,146,81,176,101,131),(67,180,117,135,82,165,102,150),(68,169,118,124,83,154,103,139),(69,158,119,143,84,173,104,128),(70,177,120,132,85,162,105,147),(71,166,91,121,86,151,106,136),(72,155,92,140,87,170,107,125),(73,174,93,129,88,159,108,144),(74,163,94,148,89,178,109,133),(75,152,95,137,90,167,110,122)], [(1,141,16,126),(2,130,17,145),(3,149,18,134),(4,138,19,123),(5,127,20,142),(6,146,21,131),(7,135,22,150),(8,124,23,139),(9,143,24,128),(10,132,25,147),(11,121,26,136),(12,140,27,125),(13,129,28,144),(14,148,29,133),(15,137,30,122),(31,161,46,176),(32,180,47,165),(33,169,48,154),(34,158,49,173),(35,177,50,162),(36,166,51,151),(37,155,52,170),(38,174,53,159),(39,163,54,178),(40,152,55,167),(41,171,56,156),(42,160,57,175),(43,179,58,164),(44,168,59,153),(45,157,60,172),(61,196,76,181),(62,185,77,200),(63,204,78,189),(64,193,79,208),(65,182,80,197),(66,201,81,186),(67,190,82,205),(68,209,83,194),(69,198,84,183),(70,187,85,202),(71,206,86,191),(72,195,87,210),(73,184,88,199),(74,203,89,188),(75,192,90,207),(91,221,106,236),(92,240,107,225),(93,229,108,214),(94,218,109,233),(95,237,110,222),(96,226,111,211),(97,215,112,230),(98,234,113,219),(99,223,114,238),(100,212,115,227),(101,231,116,216),(102,220,117,235),(103,239,118,224),(104,228,119,213),(105,217,120,232)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 20A | 20B | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 40A | 40B | 40C | 40D | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 4 | 12 | 30 | 2 | 2 | 15 | 15 | 20 | 60 | 2 | 2 | 2 | 8 | 6 | 6 | 10 | 10 | 2 | 2 | 8 | 8 | 24 | 24 | 4 | 40 | 4 | 4 | 4 | 4 | 20 | 20 | 4 | 4 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | 8 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | C4○D8 | S3×D4 | S3×D5 | D4×D5 | Q8.7D6 | C2×S3×D5 | D8⋊3D5 | D10⋊D6 | D30.11D4 |
| kernel | D30.11D4 | D15⋊2C8 | D12.D5 | C3⋊Dic20 | C3×D4.D5 | C5×D4⋊S3 | D12⋊D5 | D4⋊2D15 | D4.D5 | Dic15 | D30 | D4⋊S3 | C5⋊2C8 | Dic10 | C5×D4 | C3⋊C8 | D12 | C3×D4 | C15 | C10 | D4 | 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 | 4 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D30.11D4 ►in GL6(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 190 | 240 |
| 0 | 0 | 0 | 0 | 191 | 240 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 160 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 189 |
| 0 | 0 | 0 | 0 | 190 | 0 |
| 233 | 0 | 0 | 0 | 0 | 0 |
| 168 | 211 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 1 |
| 0 | 0 | 0 | 0 | 51 | 190 |
| 211 | 90 | 0 | 0 | 0 | 0 |
| 73 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 1 |
| 0 | 0 | 0 | 0 | 51 | 190 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,240,0,0,0,0,1,0,0,0,0,0,0,0,190,191,0,0,0,0,240,240],[240,160,0,0,0,0,0,1,0,0,0,0,0,0,240,240,0,0,0,0,0,1,0,0,0,0,0,0,0,190,0,0,0,0,189,0],[233,168,0,0,0,0,0,211,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,51,51,0,0,0,0,1,190],[211,73,0,0,0,0,90,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,51,51,0,0,0,0,1,190] >;
D30.11D4 in GAP, Magma, Sage, TeX
D_{30}._{11}D_4 % in TeX
G:=Group("D30.11D4"); // GroupNames label
G:=SmallGroup(480,575);
// by ID
G=gap.SmallGroup(480,575);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,135,675,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=1,c^4=d^2=a^15,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^19,c*b*c^-1=a^18*b,d*b*d^-1=a^3*b,d*c*d^-1=c^3>;
// generators/relations