metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.9D4, C60.12C23, Dic6.7D10, Dic10.7D6, Dic15.41D4, Dic30.3C22, C3⋊C8.6D10, D15⋊Q8⋊2C2, D4.D5⋊3S3, D4.S3⋊3D5, (C5×D4).4D6, C6.70(D4×D5), C5⋊2C8.6D6, C3⋊Dic20⋊2C2, D4.14(S3×D5), (C3×D4).4D10, C5⋊Dic12⋊2C2, C10.71(S3×D4), C5⋊3(D4.D6), C30.174(C2×D4), C3⋊3(SD16⋊D5), D30.5C4⋊2C2, D4⋊2D15.1C2, C15⋊15(C8.C22), C20.12(C22×S3), (C4×D15).4C22, (D4×C15).6C22, C12.12(C22×D5), C2.23(D10⋊D6), (C5×Dic6).3C22, (C3×Dic10).3C22, C4.12(C2×S3×D5), (C5×D4.S3)⋊4C2, (C3×D4.D5)⋊4C2, (C5×C3⋊C8).2C22, (C3×C5⋊2C8).2C22, SmallGroup(480,564)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30.9D4
G = < a,b,c,d | a30=b2=1, c4=d2=a15, bab=a-1, cac-1=dad-1=a19, cbc-1=a3b, dbd-1=a18b, dcd-1=c3 >
Subgroups: 668 in 120 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, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, D15, C30, C30, C8.C22, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×SD16, D4⋊2S3, S3×Q8, C5×Dic3, C3×Dic5, Dic15, Dic15, C60, D30, C2×C30, C8⋊D5, Dic20, D4.D5, C5⋊Q16, C5×SD16, D4⋊2D5, Q8×D5, D4.D6, C5×C3⋊C8, C3×C5⋊2C8, D30.C2, C15⋊Q8, C3×Dic10, C5×Dic6, Dic30, C4×D15, C2×Dic15, C15⋊7D4, D4×C15, SD16⋊D5, D30.5C4, C3⋊Dic20, C5⋊Dic12, C3×D4.D5, C5×D4.S3, D15⋊Q8, D4⋊2D15, D30.9D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8.C22, C22×D5, S3×D4, S3×D5, D4×D5, D4.D6, C2×S3×D5, SD16⋊D5, D10⋊D6, D30.9D4
►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 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)(91 105)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)(121 140)(122 139)(123 138)(124 137)(125 136)(126 135)(127 134)(128 133)(129 132)(130 131)(141 150)(142 149)(143 148)(144 147)(145 146)(151 170)(152 169)(153 168)(154 167)(155 166)(156 165)(157 164)(158 163)(159 162)(160 161)(171 180)(172 179)(173 178)(174 177)(175 176)(181 185)(182 184)(186 210)(187 209)(188 208)(189 207)(190 206)(191 205)(192 204)(193 203)(194 202)(195 201)(196 200)(197 199)(211 235)(212 234)(213 233)(214 232)(215 231)(216 230)(217 229)(218 228)(219 227)(220 226)(221 225)(222 224)(236 240)(237 239)
(1 216 56 206 16 231 41 191)(2 235 57 195 17 220 42 210)(3 224 58 184 18 239 43 199)(4 213 59 203 19 228 44 188)(5 232 60 192 20 217 45 207)(6 221 31 181 21 236 46 196)(7 240 32 200 22 225 47 185)(8 229 33 189 23 214 48 204)(9 218 34 208 24 233 49 193)(10 237 35 197 25 222 50 182)(11 226 36 186 26 211 51 201)(12 215 37 205 27 230 52 190)(13 234 38 194 28 219 53 209)(14 223 39 183 29 238 54 198)(15 212 40 202 30 227 55 187)(61 156 116 126 76 171 101 141)(62 175 117 145 77 160 102 130)(63 164 118 134 78 179 103 149)(64 153 119 123 79 168 104 138)(65 172 120 142 80 157 105 127)(66 161 91 131 81 176 106 146)(67 180 92 150 82 165 107 135)(68 169 93 139 83 154 108 124)(69 158 94 128 84 173 109 143)(70 177 95 147 85 162 110 132)(71 166 96 136 86 151 111 121)(72 155 97 125 87 170 112 140)(73 174 98 144 88 159 113 129)(74 163 99 133 89 178 114 148)(75 152 100 122 90 167 115 137)
(1 131 16 146)(2 150 17 135)(3 139 18 124)(4 128 19 143)(5 147 20 132)(6 136 21 121)(7 125 22 140)(8 144 23 129)(9 133 24 148)(10 122 25 137)(11 141 26 126)(12 130 27 145)(13 149 28 134)(14 138 29 123)(15 127 30 142)(31 166 46 151)(32 155 47 170)(33 174 48 159)(34 163 49 178)(35 152 50 167)(36 171 51 156)(37 160 52 175)(38 179 53 164)(39 168 54 153)(40 157 55 172)(41 176 56 161)(42 165 57 180)(43 154 58 169)(44 173 59 158)(45 162 60 177)(61 201 76 186)(62 190 77 205)(63 209 78 194)(64 198 79 183)(65 187 80 202)(66 206 81 191)(67 195 82 210)(68 184 83 199)(69 203 84 188)(70 192 85 207)(71 181 86 196)(72 200 87 185)(73 189 88 204)(74 208 89 193)(75 197 90 182)(91 216 106 231)(92 235 107 220)(93 224 108 239)(94 213 109 228)(95 232 110 217)(96 221 111 236)(97 240 112 225)(98 229 113 214)(99 218 114 233)(100 237 115 222)(101 226 116 211)(102 215 117 230)(103 234 118 219)(104 223 119 238)(105 212 120 227)
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)(61,85)(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)(91,105)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)(121,140)(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)(141,150)(142,149)(143,148)(144,147)(145,146)(151,170)(152,169)(153,168)(154,167)(155,166)(156,165)(157,164)(158,163)(159,162)(160,161)(171,180)(172,179)(173,178)(174,177)(175,176)(181,185)(182,184)(186,210)(187,209)(188,208)(189,207)(190,206)(191,205)(192,204)(193,203)(194,202)(195,201)(196,200)(197,199)(211,235)(212,234)(213,233)(214,232)(215,231)(216,230)(217,229)(218,228)(219,227)(220,226)(221,225)(222,224)(236,240)(237,239), (1,216,56,206,16,231,41,191)(2,235,57,195,17,220,42,210)(3,224,58,184,18,239,43,199)(4,213,59,203,19,228,44,188)(5,232,60,192,20,217,45,207)(6,221,31,181,21,236,46,196)(7,240,32,200,22,225,47,185)(8,229,33,189,23,214,48,204)(9,218,34,208,24,233,49,193)(10,237,35,197,25,222,50,182)(11,226,36,186,26,211,51,201)(12,215,37,205,27,230,52,190)(13,234,38,194,28,219,53,209)(14,223,39,183,29,238,54,198)(15,212,40,202,30,227,55,187)(61,156,116,126,76,171,101,141)(62,175,117,145,77,160,102,130)(63,164,118,134,78,179,103,149)(64,153,119,123,79,168,104,138)(65,172,120,142,80,157,105,127)(66,161,91,131,81,176,106,146)(67,180,92,150,82,165,107,135)(68,169,93,139,83,154,108,124)(69,158,94,128,84,173,109,143)(70,177,95,147,85,162,110,132)(71,166,96,136,86,151,111,121)(72,155,97,125,87,170,112,140)(73,174,98,144,88,159,113,129)(74,163,99,133,89,178,114,148)(75,152,100,122,90,167,115,137), (1,131,16,146)(2,150,17,135)(3,139,18,124)(4,128,19,143)(5,147,20,132)(6,136,21,121)(7,125,22,140)(8,144,23,129)(9,133,24,148)(10,122,25,137)(11,141,26,126)(12,130,27,145)(13,149,28,134)(14,138,29,123)(15,127,30,142)(31,166,46,151)(32,155,47,170)(33,174,48,159)(34,163,49,178)(35,152,50,167)(36,171,51,156)(37,160,52,175)(38,179,53,164)(39,168,54,153)(40,157,55,172)(41,176,56,161)(42,165,57,180)(43,154,58,169)(44,173,59,158)(45,162,60,177)(61,201,76,186)(62,190,77,205)(63,209,78,194)(64,198,79,183)(65,187,80,202)(66,206,81,191)(67,195,82,210)(68,184,83,199)(69,203,84,188)(70,192,85,207)(71,181,86,196)(72,200,87,185)(73,189,88,204)(74,208,89,193)(75,197,90,182)(91,216,106,231)(92,235,107,220)(93,224,108,239)(94,213,109,228)(95,232,110,217)(96,221,111,236)(97,240,112,225)(98,229,113,214)(99,218,114,233)(100,237,115,222)(101,226,116,211)(102,215,117,230)(103,234,118,219)(104,223,119,238)(105,212,120,227)>;
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)(61,85)(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)(91,105)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)(121,140)(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)(141,150)(142,149)(143,148)(144,147)(145,146)(151,170)(152,169)(153,168)(154,167)(155,166)(156,165)(157,164)(158,163)(159,162)(160,161)(171,180)(172,179)(173,178)(174,177)(175,176)(181,185)(182,184)(186,210)(187,209)(188,208)(189,207)(190,206)(191,205)(192,204)(193,203)(194,202)(195,201)(196,200)(197,199)(211,235)(212,234)(213,233)(214,232)(215,231)(216,230)(217,229)(218,228)(219,227)(220,226)(221,225)(222,224)(236,240)(237,239), (1,216,56,206,16,231,41,191)(2,235,57,195,17,220,42,210)(3,224,58,184,18,239,43,199)(4,213,59,203,19,228,44,188)(5,232,60,192,20,217,45,207)(6,221,31,181,21,236,46,196)(7,240,32,200,22,225,47,185)(8,229,33,189,23,214,48,204)(9,218,34,208,24,233,49,193)(10,237,35,197,25,222,50,182)(11,226,36,186,26,211,51,201)(12,215,37,205,27,230,52,190)(13,234,38,194,28,219,53,209)(14,223,39,183,29,238,54,198)(15,212,40,202,30,227,55,187)(61,156,116,126,76,171,101,141)(62,175,117,145,77,160,102,130)(63,164,118,134,78,179,103,149)(64,153,119,123,79,168,104,138)(65,172,120,142,80,157,105,127)(66,161,91,131,81,176,106,146)(67,180,92,150,82,165,107,135)(68,169,93,139,83,154,108,124)(69,158,94,128,84,173,109,143)(70,177,95,147,85,162,110,132)(71,166,96,136,86,151,111,121)(72,155,97,125,87,170,112,140)(73,174,98,144,88,159,113,129)(74,163,99,133,89,178,114,148)(75,152,100,122,90,167,115,137), (1,131,16,146)(2,150,17,135)(3,139,18,124)(4,128,19,143)(5,147,20,132)(6,136,21,121)(7,125,22,140)(8,144,23,129)(9,133,24,148)(10,122,25,137)(11,141,26,126)(12,130,27,145)(13,149,28,134)(14,138,29,123)(15,127,30,142)(31,166,46,151)(32,155,47,170)(33,174,48,159)(34,163,49,178)(35,152,50,167)(36,171,51,156)(37,160,52,175)(38,179,53,164)(39,168,54,153)(40,157,55,172)(41,176,56,161)(42,165,57,180)(43,154,58,169)(44,173,59,158)(45,162,60,177)(61,201,76,186)(62,190,77,205)(63,209,78,194)(64,198,79,183)(65,187,80,202)(66,206,81,191)(67,195,82,210)(68,184,83,199)(69,203,84,188)(70,192,85,207)(71,181,86,196)(72,200,87,185)(73,189,88,204)(74,208,89,193)(75,197,90,182)(91,216,106,231)(92,235,107,220)(93,224,108,239)(94,213,109,228)(95,232,110,217)(96,221,111,236)(97,240,112,225)(98,229,113,214)(99,218,114,233)(100,237,115,222)(101,226,116,211)(102,215,117,230)(103,234,118,219)(104,223,119,238)(105,212,120,227) );
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),(61,85),(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),(91,105),(92,104),(93,103),(94,102),(95,101),(96,100),(97,99),(106,120),(107,119),(108,118),(109,117),(110,116),(111,115),(112,114),(121,140),(122,139),(123,138),(124,137),(125,136),(126,135),(127,134),(128,133),(129,132),(130,131),(141,150),(142,149),(143,148),(144,147),(145,146),(151,170),(152,169),(153,168),(154,167),(155,166),(156,165),(157,164),(158,163),(159,162),(160,161),(171,180),(172,179),(173,178),(174,177),(175,176),(181,185),(182,184),(186,210),(187,209),(188,208),(189,207),(190,206),(191,205),(192,204),(193,203),(194,202),(195,201),(196,200),(197,199),(211,235),(212,234),(213,233),(214,232),(215,231),(216,230),(217,229),(218,228),(219,227),(220,226),(221,225),(222,224),(236,240),(237,239)], [(1,216,56,206,16,231,41,191),(2,235,57,195,17,220,42,210),(3,224,58,184,18,239,43,199),(4,213,59,203,19,228,44,188),(5,232,60,192,20,217,45,207),(6,221,31,181,21,236,46,196),(7,240,32,200,22,225,47,185),(8,229,33,189,23,214,48,204),(9,218,34,208,24,233,49,193),(10,237,35,197,25,222,50,182),(11,226,36,186,26,211,51,201),(12,215,37,205,27,230,52,190),(13,234,38,194,28,219,53,209),(14,223,39,183,29,238,54,198),(15,212,40,202,30,227,55,187),(61,156,116,126,76,171,101,141),(62,175,117,145,77,160,102,130),(63,164,118,134,78,179,103,149),(64,153,119,123,79,168,104,138),(65,172,120,142,80,157,105,127),(66,161,91,131,81,176,106,146),(67,180,92,150,82,165,107,135),(68,169,93,139,83,154,108,124),(69,158,94,128,84,173,109,143),(70,177,95,147,85,162,110,132),(71,166,96,136,86,151,111,121),(72,155,97,125,87,170,112,140),(73,174,98,144,88,159,113,129),(74,163,99,133,89,178,114,148),(75,152,100,122,90,167,115,137)], [(1,131,16,146),(2,150,17,135),(3,139,18,124),(4,128,19,143),(5,147,20,132),(6,136,21,121),(7,125,22,140),(8,144,23,129),(9,133,24,148),(10,122,25,137),(11,141,26,126),(12,130,27,145),(13,149,28,134),(14,138,29,123),(15,127,30,142),(31,166,46,151),(32,155,47,170),(33,174,48,159),(34,163,49,178),(35,152,50,167),(36,171,51,156),(37,160,52,175),(38,179,53,164),(39,168,54,153),(40,157,55,172),(41,176,56,161),(42,165,57,180),(43,154,58,169),(44,173,59,158),(45,162,60,177),(61,201,76,186),(62,190,77,205),(63,209,78,194),(64,198,79,183),(65,187,80,202),(66,206,81,191),(67,195,82,210),(68,184,83,199),(69,203,84,188),(70,192,85,207),(71,181,86,196),(72,200,87,185),(73,189,88,204),(74,208,89,193),(75,197,90,182),(91,216,106,231),(92,235,107,220),(93,224,108,239),(94,213,109,228),(95,232,110,217),(96,221,111,236),(97,240,112,225),(98,229,113,214),(99,218,114,233),(100,237,115,222),(101,226,116,211),(102,215,117,230),(103,234,118,219),(104,223,119,238),(105,212,120,227)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 40A | 40B | 40C | 40D | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 4 | 30 | 2 | 2 | 12 | 20 | 30 | 60 | 2 | 2 | 2 | 8 | 12 | 20 | 2 | 2 | 8 | 8 | 4 | 40 | 4 | 4 | 4 | 4 | 24 | 24 | 20 | 20 | 4 | 4 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | 8 |
42 irreducible representations
| dim | 1 | 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 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C8.C22 | S3×D4 | S3×D5 | D4×D5 | D4.D6 | C2×S3×D5 | SD16⋊D5 | D10⋊D6 | D30.9D4 |
| kernel | D30.9D4 | D30.5C4 | C3⋊Dic20 | C5⋊Dic12 | C3×D4.D5 | C5×D4.S3 | D15⋊Q8 | D4⋊2D15 | D4.D5 | Dic15 | D30 | D4.S3 | C5⋊2C8 | Dic10 | C5×D4 | C3⋊C8 | Dic6 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D30.9D4 ►in GL8(𝔽241)
| 52 | 1 | 189 | 240 | 0 | 0 | 0 | 0 |
| 240 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 52 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 52 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 52 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 189 | 189 | 0 | 0 | 0 | 0 | 0 | 0 |
| 52 | 1 | 189 | 240 | 0 | 0 | 0 | 0 |
| 189 | 189 | 52 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 189 | 189 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 52 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 189 | 189 |
| 122 | 79 | 230 | 83 | 0 | 0 | 0 | 0 |
| 1 | 119 | 173 | 11 | 0 | 0 | 0 | 0 |
| 11 | 158 | 111 | 162 | 0 | 0 | 0 | 0 |
| 68 | 230 | 174 | 130 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 29 | 31 | 29 |
| 0 | 0 | 0 | 0 | 104 | 210 | 104 | 210 |
| 0 | 0 | 0 | 0 | 210 | 212 | 31 | 29 |
| 0 | 0 | 0 | 0 | 137 | 31 | 104 | 210 |
| 122 | 79 | 230 | 83 | 0 | 0 | 0 | 0 |
| 1 | 119 | 173 | 11 | 0 | 0 | 0 | 0 |
| 11 | 158 | 111 | 162 | 0 | 0 | 0 | 0 |
| 68 | 230 | 174 | 130 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 96 | 0 | 172 | 0 |
| 0 | 0 | 0 | 0 | 69 | 145 | 214 | 69 |
| 0 | 0 | 0 | 0 | 172 | 0 | 145 | 0 |
| 0 | 0 | 0 | 0 | 214 | 69 | 172 | 96 |
G:=sub<GL(8,GF(241))| [52,240,52,240,0,0,0,0,1,0,1,0,0,0,0,0,189,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,0,52,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,52,240,0,0,0,0,0,0,1,0],[52,189,52,189,0,0,0,0,1,189,1,189,0,0,0,0,0,0,189,52,0,0,0,0,0,0,240,52,0,0,0,0,0,0,0,0,52,189,0,0,0,0,0,0,1,189,0,0,0,0,0,0,0,0,52,189,0,0,0,0,0,0,1,189],[122,1,11,68,0,0,0,0,79,119,158,230,0,0,0,0,230,173,111,174,0,0,0,0,83,11,162,130,0,0,0,0,0,0,0,0,31,104,210,137,0,0,0,0,29,210,212,31,0,0,0,0,31,104,31,104,0,0,0,0,29,210,29,210],[122,1,11,68,0,0,0,0,79,119,158,230,0,0,0,0,230,173,111,174,0,0,0,0,83,11,162,130,0,0,0,0,0,0,0,0,96,69,172,214,0,0,0,0,0,145,0,69,0,0,0,0,172,214,145,172,0,0,0,0,0,69,0,96] >;
D30.9D4 in GAP, Magma, Sage, TeX
D_{30}._9D_4 % in TeX
G:=Group("D30.9D4"); // GroupNames label
G:=SmallGroup(480,564);
// by ID
G=gap.SmallGroup(480,564);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,303,675,346,185,80,1356,18822]);
// 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^3*b,d*b*d^-1=a^18*b,d*c*d^-1=c^3>;
// generators/relations