metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.14D4, D20.16D6, C60.45C23, Dic6.16D10, Dic15.46D4, D60.17C22, Q8⋊D5⋊5S3, C3⋊D40⋊8C2, C3⋊C8.21D10, C3⋊Q16⋊5D5, C6.79(D4×D5), C15⋊23(C4○D8), C10.80(S3×D4), C5⋊2C8.21D6, (C5×Q8).26D6, (C3×Q8).9D10, Q8.22(S3×D5), D15⋊2C8⋊6C2, D20⋊S3⋊3C2, Q8⋊3D15⋊2C2, C3⋊3(Q8.D10), Dic6⋊D5⋊8C2, C30.207(C2×D4), C5⋊4(Q8.7D6), C20.45(C22×S3), C12.45(C22×D5), (C4×D15).13C22, (C3×D20).17C22, (Q8×C15).15C22, C2.32(D10⋊D6), (C5×Dic6).17C22, C4.45(C2×S3×D5), (C3×Q8⋊D5)⋊7C2, (C5×C3⋊Q16)⋊7C2, (C5×C3⋊C8).15C22, (C3×C5⋊2C8).15C22, SmallGroup(480,597)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30.14D4
G = < a,b,c,d | a30=b2=d2=1, c4=a15, bab=a-1, cac-1=dad=a19, cbc-1=a18b, dbd=a3b, dcd=c3 >
Subgroups: 796 in 124 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×2], C6, C6, C8 [×2], C2×C4 [×3], D4 [×4], Q8, Q8, D5 [×3], C10, Dic3 [×2], C12, C12, D6 [×2], C2×C6, C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, C20, C20 [×2], D10 [×3], C3⋊C8, C24, Dic6, C4×S3 [×2], D12 [×2], C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C3×D5, D15 [×2], C30, C4○D8, C5⋊2C8, C40, C4×D5 [×3], D20, D20 [×3], C5×Q8, C5×Q8, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C3×SD16, D4⋊2S3, Q8⋊3S3, C5×Dic3, Dic15, C60, C60, C6×D5, D30, D30, C8×D5, D40, Q8⋊D5, Q8⋊D5, C5×Q16, Q8⋊2D5 [×2], Q8.7D6, C5×C3⋊C8, C3×C5⋊2C8, D5×Dic3, C3⋊D20, C3×D20, C5×Dic6, C4×D15, C4×D15, D60, D60, Q8×C15, Q8.D10, D15⋊2C8, C3⋊D40, Dic6⋊D5, C3×Q8⋊D5, C5×C3⋊Q16, D20⋊S3, Q8⋊3D15, D30.14D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C4○D8, C22×D5, S3×D4, S3×D5, D4×D5, Q8.7D6, C2×S3×D5, Q8.D10, D10⋊D6, D30.14D4
►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)(91 105)(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)(121 125)(122 124)(126 150)(127 149)(128 148)(129 147)(130 146)(131 145)(132 144)(133 143)(134 142)(135 141)(136 140)(137 139)(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)(181 200)(182 199)(183 198)(184 197)(185 196)(186 195)(187 194)(188 193)(189 192)(190 191)(201 210)(202 209)(203 208)(204 207)(205 206)(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)
(1 226 56 206 16 211 41 191)(2 215 57 195 17 230 42 210)(3 234 58 184 18 219 43 199)(4 223 59 203 19 238 44 188)(5 212 60 192 20 227 45 207)(6 231 31 181 21 216 46 196)(7 220 32 200 22 235 47 185)(8 239 33 189 23 224 48 204)(9 228 34 208 24 213 49 193)(10 217 35 197 25 232 50 182)(11 236 36 186 26 221 51 201)(12 225 37 205 27 240 52 190)(13 214 38 194 28 229 53 209)(14 233 39 183 29 218 54 198)(15 222 40 202 30 237 55 187)(61 156 91 146 76 171 106 131)(62 175 92 135 77 160 107 150)(63 164 93 124 78 179 108 139)(64 153 94 143 79 168 109 128)(65 172 95 132 80 157 110 147)(66 161 96 121 81 176 111 136)(67 180 97 140 82 165 112 125)(68 169 98 129 83 154 113 144)(69 158 99 148 84 173 114 133)(70 177 100 137 85 162 115 122)(71 166 101 126 86 151 116 141)(72 155 102 145 87 170 117 130)(73 174 103 134 88 159 118 149)(74 163 104 123 89 178 119 138)(75 152 105 142 90 167 120 127)
(1 131)(2 150)(3 139)(4 128)(5 147)(6 136)(7 125)(8 144)(9 133)(10 122)(11 141)(12 130)(13 149)(14 138)(15 127)(16 146)(17 135)(18 124)(19 143)(20 132)(21 121)(22 140)(23 129)(24 148)(25 137)(26 126)(27 145)(28 134)(29 123)(30 142)(31 176)(32 165)(33 154)(34 173)(35 162)(36 151)(37 170)(38 159)(39 178)(40 167)(41 156)(42 175)(43 164)(44 153)(45 172)(46 161)(47 180)(48 169)(49 158)(50 177)(51 166)(52 155)(53 174)(54 163)(55 152)(56 171)(57 160)(58 179)(59 168)(60 157)(61 206)(62 195)(63 184)(64 203)(65 192)(66 181)(67 200)(68 189)(69 208)(70 197)(71 186)(72 205)(73 194)(74 183)(75 202)(76 191)(77 210)(78 199)(79 188)(80 207)(81 196)(82 185)(83 204)(84 193)(85 182)(86 201)(87 190)(88 209)(89 198)(90 187)(91 226)(92 215)(93 234)(94 223)(95 212)(96 231)(97 220)(98 239)(99 228)(100 217)(101 236)(102 225)(103 214)(104 233)(105 222)(106 211)(107 230)(108 219)(109 238)(110 227)(111 216)(112 235)(113 224)(114 213)(115 232)(116 221)(117 240)(118 229)(119 218)(120 237)
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)(91,105)(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)(121,125)(122,124)(126,150)(127,149)(128,148)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(136,140)(137,139)(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)(181,200)(182,199)(183,198)(184,197)(185,196)(186,195)(187,194)(188,193)(189,192)(190,191)(201,210)(202,209)(203,208)(204,207)(205,206)(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), (1,226,56,206,16,211,41,191)(2,215,57,195,17,230,42,210)(3,234,58,184,18,219,43,199)(4,223,59,203,19,238,44,188)(5,212,60,192,20,227,45,207)(6,231,31,181,21,216,46,196)(7,220,32,200,22,235,47,185)(8,239,33,189,23,224,48,204)(9,228,34,208,24,213,49,193)(10,217,35,197,25,232,50,182)(11,236,36,186,26,221,51,201)(12,225,37,205,27,240,52,190)(13,214,38,194,28,229,53,209)(14,233,39,183,29,218,54,198)(15,222,40,202,30,237,55,187)(61,156,91,146,76,171,106,131)(62,175,92,135,77,160,107,150)(63,164,93,124,78,179,108,139)(64,153,94,143,79,168,109,128)(65,172,95,132,80,157,110,147)(66,161,96,121,81,176,111,136)(67,180,97,140,82,165,112,125)(68,169,98,129,83,154,113,144)(69,158,99,148,84,173,114,133)(70,177,100,137,85,162,115,122)(71,166,101,126,86,151,116,141)(72,155,102,145,87,170,117,130)(73,174,103,134,88,159,118,149)(74,163,104,123,89,178,119,138)(75,152,105,142,90,167,120,127), (1,131)(2,150)(3,139)(4,128)(5,147)(6,136)(7,125)(8,144)(9,133)(10,122)(11,141)(12,130)(13,149)(14,138)(15,127)(16,146)(17,135)(18,124)(19,143)(20,132)(21,121)(22,140)(23,129)(24,148)(25,137)(26,126)(27,145)(28,134)(29,123)(30,142)(31,176)(32,165)(33,154)(34,173)(35,162)(36,151)(37,170)(38,159)(39,178)(40,167)(41,156)(42,175)(43,164)(44,153)(45,172)(46,161)(47,180)(48,169)(49,158)(50,177)(51,166)(52,155)(53,174)(54,163)(55,152)(56,171)(57,160)(58,179)(59,168)(60,157)(61,206)(62,195)(63,184)(64,203)(65,192)(66,181)(67,200)(68,189)(69,208)(70,197)(71,186)(72,205)(73,194)(74,183)(75,202)(76,191)(77,210)(78,199)(79,188)(80,207)(81,196)(82,185)(83,204)(84,193)(85,182)(86,201)(87,190)(88,209)(89,198)(90,187)(91,226)(92,215)(93,234)(94,223)(95,212)(96,231)(97,220)(98,239)(99,228)(100,217)(101,236)(102,225)(103,214)(104,233)(105,222)(106,211)(107,230)(108,219)(109,238)(110,227)(111,216)(112,235)(113,224)(114,213)(115,232)(116,221)(117,240)(118,229)(119,218)(120,237)>;
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)(91,105)(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)(121,125)(122,124)(126,150)(127,149)(128,148)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(136,140)(137,139)(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)(181,200)(182,199)(183,198)(184,197)(185,196)(186,195)(187,194)(188,193)(189,192)(190,191)(201,210)(202,209)(203,208)(204,207)(205,206)(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), (1,226,56,206,16,211,41,191)(2,215,57,195,17,230,42,210)(3,234,58,184,18,219,43,199)(4,223,59,203,19,238,44,188)(5,212,60,192,20,227,45,207)(6,231,31,181,21,216,46,196)(7,220,32,200,22,235,47,185)(8,239,33,189,23,224,48,204)(9,228,34,208,24,213,49,193)(10,217,35,197,25,232,50,182)(11,236,36,186,26,221,51,201)(12,225,37,205,27,240,52,190)(13,214,38,194,28,229,53,209)(14,233,39,183,29,218,54,198)(15,222,40,202,30,237,55,187)(61,156,91,146,76,171,106,131)(62,175,92,135,77,160,107,150)(63,164,93,124,78,179,108,139)(64,153,94,143,79,168,109,128)(65,172,95,132,80,157,110,147)(66,161,96,121,81,176,111,136)(67,180,97,140,82,165,112,125)(68,169,98,129,83,154,113,144)(69,158,99,148,84,173,114,133)(70,177,100,137,85,162,115,122)(71,166,101,126,86,151,116,141)(72,155,102,145,87,170,117,130)(73,174,103,134,88,159,118,149)(74,163,104,123,89,178,119,138)(75,152,105,142,90,167,120,127), (1,131)(2,150)(3,139)(4,128)(5,147)(6,136)(7,125)(8,144)(9,133)(10,122)(11,141)(12,130)(13,149)(14,138)(15,127)(16,146)(17,135)(18,124)(19,143)(20,132)(21,121)(22,140)(23,129)(24,148)(25,137)(26,126)(27,145)(28,134)(29,123)(30,142)(31,176)(32,165)(33,154)(34,173)(35,162)(36,151)(37,170)(38,159)(39,178)(40,167)(41,156)(42,175)(43,164)(44,153)(45,172)(46,161)(47,180)(48,169)(49,158)(50,177)(51,166)(52,155)(53,174)(54,163)(55,152)(56,171)(57,160)(58,179)(59,168)(60,157)(61,206)(62,195)(63,184)(64,203)(65,192)(66,181)(67,200)(68,189)(69,208)(70,197)(71,186)(72,205)(73,194)(74,183)(75,202)(76,191)(77,210)(78,199)(79,188)(80,207)(81,196)(82,185)(83,204)(84,193)(85,182)(86,201)(87,190)(88,209)(89,198)(90,187)(91,226)(92,215)(93,234)(94,223)(95,212)(96,231)(97,220)(98,239)(99,228)(100,217)(101,236)(102,225)(103,214)(104,233)(105,222)(106,211)(107,230)(108,219)(109,238)(110,227)(111,216)(112,235)(113,224)(114,213)(115,232)(116,221)(117,240)(118,229)(119,218)(120,237) );
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),(91,105),(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),(121,125),(122,124),(126,150),(127,149),(128,148),(129,147),(130,146),(131,145),(132,144),(133,143),(134,142),(135,141),(136,140),(137,139),(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),(181,200),(182,199),(183,198),(184,197),(185,196),(186,195),(187,194),(188,193),(189,192),(190,191),(201,210),(202,209),(203,208),(204,207),(205,206),(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)], [(1,226,56,206,16,211,41,191),(2,215,57,195,17,230,42,210),(3,234,58,184,18,219,43,199),(4,223,59,203,19,238,44,188),(5,212,60,192,20,227,45,207),(6,231,31,181,21,216,46,196),(7,220,32,200,22,235,47,185),(8,239,33,189,23,224,48,204),(9,228,34,208,24,213,49,193),(10,217,35,197,25,232,50,182),(11,236,36,186,26,221,51,201),(12,225,37,205,27,240,52,190),(13,214,38,194,28,229,53,209),(14,233,39,183,29,218,54,198),(15,222,40,202,30,237,55,187),(61,156,91,146,76,171,106,131),(62,175,92,135,77,160,107,150),(63,164,93,124,78,179,108,139),(64,153,94,143,79,168,109,128),(65,172,95,132,80,157,110,147),(66,161,96,121,81,176,111,136),(67,180,97,140,82,165,112,125),(68,169,98,129,83,154,113,144),(69,158,99,148,84,173,114,133),(70,177,100,137,85,162,115,122),(71,166,101,126,86,151,116,141),(72,155,102,145,87,170,117,130),(73,174,103,134,88,159,118,149),(74,163,104,123,89,178,119,138),(75,152,105,142,90,167,120,127)], [(1,131),(2,150),(3,139),(4,128),(5,147),(6,136),(7,125),(8,144),(9,133),(10,122),(11,141),(12,130),(13,149),(14,138),(15,127),(16,146),(17,135),(18,124),(19,143),(20,132),(21,121),(22,140),(23,129),(24,148),(25,137),(26,126),(27,145),(28,134),(29,123),(30,142),(31,176),(32,165),(33,154),(34,173),(35,162),(36,151),(37,170),(38,159),(39,178),(40,167),(41,156),(42,175),(43,164),(44,153),(45,172),(46,161),(47,180),(48,169),(49,158),(50,177),(51,166),(52,155),(53,174),(54,163),(55,152),(56,171),(57,160),(58,179),(59,168),(60,157),(61,206),(62,195),(63,184),(64,203),(65,192),(66,181),(67,200),(68,189),(69,208),(70,197),(71,186),(72,205),(73,194),(74,183),(75,202),(76,191),(77,210),(78,199),(79,188),(80,207),(81,196),(82,185),(83,204),(84,193),(85,182),(86,201),(87,190),(88,209),(89,198),(90,187),(91,226),(92,215),(93,234),(94,223),(95,212),(96,231),(97,220),(98,239),(99,228),(100,217),(101,236),(102,225),(103,214),(104,233),(105,222),(106,211),(107,230),(108,219),(109,238),(110,227),(111,216),(112,235),(113,224),(114,213),(115,232),(116,221),(117,240),(118,229),(119,218),(120,237)])
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 | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 30A | 30B | 40A | 40B | 40C | 40D | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 20 | 30 | 60 | 2 | 2 | 4 | 12 | 15 | 15 | 2 | 2 | 2 | 40 | 6 | 6 | 10 | 10 | 2 | 2 | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 24 | 24 | 20 | 20 | 4 | 4 | 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 | Q8.D10 | D10⋊D6 | D30.14D4 |
| kernel | D30.14D4 | D15⋊2C8 | C3⋊D40 | Dic6⋊D5 | C3×Q8⋊D5 | C5×C3⋊Q16 | D20⋊S3 | Q8⋊3D15 | Q8⋊D5 | Dic15 | D30 | C3⋊Q16 | C5⋊2C8 | D20 | C5×Q8 | C3⋊C8 | Dic6 | C3×Q8 | C15 | C10 | Q8 | 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 D30.14D4 ►in GL6(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 189 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 239 | 192 |
| 0 | 0 | 0 | 0 | 64 | 1 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 189 | 240 | 0 | 0 |
| 0 | 0 | 52 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 64 | 1 |
| 30 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 189 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 211 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 189 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,189,1,0,0,0,0,240,0,0,0,0,0,0,0,239,64,0,0,0,0,192,1],[240,0,0,0,0,0,0,1,0,0,0,0,0,0,189,52,0,0,0,0,240,52,0,0,0,0,0,0,240,64,0,0,0,0,0,1],[30,0,0,0,0,0,0,8,0,0,0,0,0,0,1,189,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[0,211,0,0,0,0,8,0,0,0,0,0,0,0,1,189,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
D30.14D4 in GAP, Magma, Sage, TeX
D_{30}._{14}D_4 % in TeX
G:=Group("D30.14D4"); // GroupNames label
G:=SmallGroup(480,597);
// by ID
G=gap.SmallGroup(480,597);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,135,100,675,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=d^2=1,c^4=a^15,b*a*b=a^-1,c*a*c^-1=d*a*d=a^19,c*b*c^-1=a^18*b,d*b*d=a^3*b,d*c*d=c^3>;
// generators/relations