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G = D302Q8order 480 = 25·3·5

2nd semidirect product of D30 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D302Q8, Dic5.19D12, C4⋊Dic37D5, C6.20(D4×D5), C6.14(Q8×D5), C51(C4.D12), C2.21(D5×D12), C30.44(C2×D4), C30.41(C2×Q8), C10.14(S3×Q8), C32(D10⋊Q8), C10.20(C2×D12), (C2×C12).16D10, (C2×C20).225D6, C1513(C22⋊Q8), C6.72(C4○D20), C10.D417S3, C2.16(D15⋊Q8), C30.Q822C2, (C3×Dic5).10D4, (C2×Dic5).32D6, D303C4.10C2, D304C4.12C2, C30.119(C4○D4), (C2×C30).109C23, (C2×C60).318C22, (C2×Dic3).34D10, C10.46(D42S3), (C6×Dic5).63C22, C2.18(Dic5.D6), (C2×Dic15).88C22, (C10×Dic3).67C22, (C22×D15).36C22, (C2×C15⋊Q8)⋊7C2, (C2×C4).45(S3×D5), (C5×C4⋊Dic3)⋊18C2, C22.175(C2×S3×D5), (C2×D30.C2).5C2, (C3×C10.D4)⋊20C2, (C2×C6).121(C22×D5), (C2×C10).121(C22×S3), SmallGroup(480,495)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D302Q8
C1C5C15C30C2×C30C6×Dic5C2×D30.C2 — D302Q8
C15C2×C30 — D302Q8
C1C22C2×C4

Generators and relations for D302Q8
 G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=cac-1=a-1, dad-1=a19, cbc-1=a13b, dbd-1=a18b, dcd-1=c-1 >

Subgroups: 844 in 148 conjugacy classes, 50 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×7], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×7], Q8 [×2], C23, D5 [×2], C10 [×3], Dic3 [×3], C12 [×4], D6 [×4], C2×C6, C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×2], Dic5 [×2], C20 [×3], D10 [×4], C2×C10, Dic6 [×2], C4×S3 [×2], C2×Dic3 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, D15 [×2], C30 [×3], C22⋊Q8, Dic10 [×2], C4×D5 [×2], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C4⋊Dic3, C4⋊Dic3, D6⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4, C5×Dic3 [×2], C3×Dic5 [×2], C3×Dic5, Dic15, C60, D30 [×2], D30 [×2], C2×C30, C10.D4, C10.D4, D10⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C4.D12, D30.C2 [×2], C15⋊Q8 [×2], C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15, C2×C60, C22×D15, D10⋊Q8, D304C4, C30.Q8, C3×C10.D4, C5×C4⋊Dic3, D303C4, C2×D30.C2, C2×C15⋊Q8, D302Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D5, D6 [×3], C2×D4, C2×Q8, C4○D4, D10 [×3], D12 [×2], C22×S3, C22⋊Q8, C22×D5, C2×D12, D42S3, S3×Q8, S3×D5, C4○D20, D4×D5, Q8×D5, C4.D12, C2×S3×D5, D10⋊Q8, D15⋊Q8, D5×D12, Dic5.D6, D302Q8

Smallest permutation representation of D302Q8
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 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(56 60)(57 59)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 84)(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)(121 147)(122 146)(123 145)(124 144)(125 143)(126 142)(127 141)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)(148 150)(151 168)(152 167)(153 166)(154 165)(155 164)(156 163)(157 162)(158 161)(159 160)(169 180)(170 179)(171 178)(172 177)(173 176)(174 175)(181 197)(182 196)(183 195)(184 194)(185 193)(186 192)(187 191)(188 190)(198 210)(199 209)(200 208)(201 207)(202 206)(203 205)(211 239)(212 238)(213 237)(214 236)(215 235)(216 234)(217 233)(218 232)(219 231)(220 230)(221 229)(222 228)(223 227)(224 226)
(1 218 160 36)(2 217 161 35)(3 216 162 34)(4 215 163 33)(5 214 164 32)(6 213 165 31)(7 212 166 60)(8 211 167 59)(9 240 168 58)(10 239 169 57)(11 238 170 56)(12 237 171 55)(13 236 172 54)(14 235 173 53)(15 234 174 52)(16 233 175 51)(17 232 176 50)(18 231 177 49)(19 230 178 48)(20 229 179 47)(21 228 180 46)(22 227 151 45)(23 226 152 44)(24 225 153 43)(25 224 154 42)(26 223 155 41)(27 222 156 40)(28 221 157 39)(29 220 158 38)(30 219 159 37)(61 205 109 135)(62 204 110 134)(63 203 111 133)(64 202 112 132)(65 201 113 131)(66 200 114 130)(67 199 115 129)(68 198 116 128)(69 197 117 127)(70 196 118 126)(71 195 119 125)(72 194 120 124)(73 193 91 123)(74 192 92 122)(75 191 93 121)(76 190 94 150)(77 189 95 149)(78 188 96 148)(79 187 97 147)(80 186 98 146)(81 185 99 145)(82 184 100 144)(83 183 101 143)(84 182 102 142)(85 181 103 141)(86 210 104 140)(87 209 105 139)(88 208 106 138)(89 207 107 137)(90 206 108 136)
(1 69 160 117)(2 88 161 106)(3 77 162 95)(4 66 163 114)(5 85 164 103)(6 74 165 92)(7 63 166 111)(8 82 167 100)(9 71 168 119)(10 90 169 108)(11 79 170 97)(12 68 171 116)(13 87 172 105)(14 76 173 94)(15 65 174 113)(16 84 175 102)(17 73 176 91)(18 62 177 110)(19 81 178 99)(20 70 179 118)(21 89 180 107)(22 78 151 96)(23 67 152 115)(24 86 153 104)(25 75 154 93)(26 64 155 112)(27 83 156 101)(28 72 157 120)(29 61 158 109)(30 80 159 98)(31 192 213 122)(32 181 214 141)(33 200 215 130)(34 189 216 149)(35 208 217 138)(36 197 218 127)(37 186 219 146)(38 205 220 135)(39 194 221 124)(40 183 222 143)(41 202 223 132)(42 191 224 121)(43 210 225 140)(44 199 226 129)(45 188 227 148)(46 207 228 137)(47 196 229 126)(48 185 230 145)(49 204 231 134)(50 193 232 123)(51 182 233 142)(52 201 234 131)(53 190 235 150)(54 209 236 139)(55 198 237 128)(56 187 238 147)(57 206 239 136)(58 195 240 125)(59 184 211 144)(60 203 212 133)

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)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(56,60)(57,59)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)(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)(121,147)(122,146)(123,145)(124,144)(125,143)(126,142)(127,141)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(148,150)(151,168)(152,167)(153,166)(154,165)(155,164)(156,163)(157,162)(158,161)(159,160)(169,180)(170,179)(171,178)(172,177)(173,176)(174,175)(181,197)(182,196)(183,195)(184,194)(185,193)(186,192)(187,191)(188,190)(198,210)(199,209)(200,208)(201,207)(202,206)(203,205)(211,239)(212,238)(213,237)(214,236)(215,235)(216,234)(217,233)(218,232)(219,231)(220,230)(221,229)(222,228)(223,227)(224,226), (1,218,160,36)(2,217,161,35)(3,216,162,34)(4,215,163,33)(5,214,164,32)(6,213,165,31)(7,212,166,60)(8,211,167,59)(9,240,168,58)(10,239,169,57)(11,238,170,56)(12,237,171,55)(13,236,172,54)(14,235,173,53)(15,234,174,52)(16,233,175,51)(17,232,176,50)(18,231,177,49)(19,230,178,48)(20,229,179,47)(21,228,180,46)(22,227,151,45)(23,226,152,44)(24,225,153,43)(25,224,154,42)(26,223,155,41)(27,222,156,40)(28,221,157,39)(29,220,158,38)(30,219,159,37)(61,205,109,135)(62,204,110,134)(63,203,111,133)(64,202,112,132)(65,201,113,131)(66,200,114,130)(67,199,115,129)(68,198,116,128)(69,197,117,127)(70,196,118,126)(71,195,119,125)(72,194,120,124)(73,193,91,123)(74,192,92,122)(75,191,93,121)(76,190,94,150)(77,189,95,149)(78,188,96,148)(79,187,97,147)(80,186,98,146)(81,185,99,145)(82,184,100,144)(83,183,101,143)(84,182,102,142)(85,181,103,141)(86,210,104,140)(87,209,105,139)(88,208,106,138)(89,207,107,137)(90,206,108,136), (1,69,160,117)(2,88,161,106)(3,77,162,95)(4,66,163,114)(5,85,164,103)(6,74,165,92)(7,63,166,111)(8,82,167,100)(9,71,168,119)(10,90,169,108)(11,79,170,97)(12,68,171,116)(13,87,172,105)(14,76,173,94)(15,65,174,113)(16,84,175,102)(17,73,176,91)(18,62,177,110)(19,81,178,99)(20,70,179,118)(21,89,180,107)(22,78,151,96)(23,67,152,115)(24,86,153,104)(25,75,154,93)(26,64,155,112)(27,83,156,101)(28,72,157,120)(29,61,158,109)(30,80,159,98)(31,192,213,122)(32,181,214,141)(33,200,215,130)(34,189,216,149)(35,208,217,138)(36,197,218,127)(37,186,219,146)(38,205,220,135)(39,194,221,124)(40,183,222,143)(41,202,223,132)(42,191,224,121)(43,210,225,140)(44,199,226,129)(45,188,227,148)(46,207,228,137)(47,196,229,126)(48,185,230,145)(49,204,231,134)(50,193,232,123)(51,182,233,142)(52,201,234,131)(53,190,235,150)(54,209,236,139)(55,198,237,128)(56,187,238,147)(57,206,239,136)(58,195,240,125)(59,184,211,144)(60,203,212,133)>;

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)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(56,60)(57,59)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)(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)(121,147)(122,146)(123,145)(124,144)(125,143)(126,142)(127,141)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(148,150)(151,168)(152,167)(153,166)(154,165)(155,164)(156,163)(157,162)(158,161)(159,160)(169,180)(170,179)(171,178)(172,177)(173,176)(174,175)(181,197)(182,196)(183,195)(184,194)(185,193)(186,192)(187,191)(188,190)(198,210)(199,209)(200,208)(201,207)(202,206)(203,205)(211,239)(212,238)(213,237)(214,236)(215,235)(216,234)(217,233)(218,232)(219,231)(220,230)(221,229)(222,228)(223,227)(224,226), (1,218,160,36)(2,217,161,35)(3,216,162,34)(4,215,163,33)(5,214,164,32)(6,213,165,31)(7,212,166,60)(8,211,167,59)(9,240,168,58)(10,239,169,57)(11,238,170,56)(12,237,171,55)(13,236,172,54)(14,235,173,53)(15,234,174,52)(16,233,175,51)(17,232,176,50)(18,231,177,49)(19,230,178,48)(20,229,179,47)(21,228,180,46)(22,227,151,45)(23,226,152,44)(24,225,153,43)(25,224,154,42)(26,223,155,41)(27,222,156,40)(28,221,157,39)(29,220,158,38)(30,219,159,37)(61,205,109,135)(62,204,110,134)(63,203,111,133)(64,202,112,132)(65,201,113,131)(66,200,114,130)(67,199,115,129)(68,198,116,128)(69,197,117,127)(70,196,118,126)(71,195,119,125)(72,194,120,124)(73,193,91,123)(74,192,92,122)(75,191,93,121)(76,190,94,150)(77,189,95,149)(78,188,96,148)(79,187,97,147)(80,186,98,146)(81,185,99,145)(82,184,100,144)(83,183,101,143)(84,182,102,142)(85,181,103,141)(86,210,104,140)(87,209,105,139)(88,208,106,138)(89,207,107,137)(90,206,108,136), (1,69,160,117)(2,88,161,106)(3,77,162,95)(4,66,163,114)(5,85,164,103)(6,74,165,92)(7,63,166,111)(8,82,167,100)(9,71,168,119)(10,90,169,108)(11,79,170,97)(12,68,171,116)(13,87,172,105)(14,76,173,94)(15,65,174,113)(16,84,175,102)(17,73,176,91)(18,62,177,110)(19,81,178,99)(20,70,179,118)(21,89,180,107)(22,78,151,96)(23,67,152,115)(24,86,153,104)(25,75,154,93)(26,64,155,112)(27,83,156,101)(28,72,157,120)(29,61,158,109)(30,80,159,98)(31,192,213,122)(32,181,214,141)(33,200,215,130)(34,189,216,149)(35,208,217,138)(36,197,218,127)(37,186,219,146)(38,205,220,135)(39,194,221,124)(40,183,222,143)(41,202,223,132)(42,191,224,121)(43,210,225,140)(44,199,226,129)(45,188,227,148)(46,207,228,137)(47,196,229,126)(48,185,230,145)(49,204,231,134)(50,193,232,123)(51,182,233,142)(52,201,234,131)(53,190,235,150)(54,209,236,139)(55,198,237,128)(56,187,238,147)(57,206,239,136)(58,195,240,125)(59,184,211,144)(60,203,212,133) );

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),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(37,49),(38,48),(39,47),(40,46),(41,45),(42,44),(56,60),(57,59),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,70),(68,69),(77,90),(78,89),(79,88),(80,87),(81,86),(82,85),(83,84),(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),(121,147),(122,146),(123,145),(124,144),(125,143),(126,142),(127,141),(128,140),(129,139),(130,138),(131,137),(132,136),(133,135),(148,150),(151,168),(152,167),(153,166),(154,165),(155,164),(156,163),(157,162),(158,161),(159,160),(169,180),(170,179),(171,178),(172,177),(173,176),(174,175),(181,197),(182,196),(183,195),(184,194),(185,193),(186,192),(187,191),(188,190),(198,210),(199,209),(200,208),(201,207),(202,206),(203,205),(211,239),(212,238),(213,237),(214,236),(215,235),(216,234),(217,233),(218,232),(219,231),(220,230),(221,229),(222,228),(223,227),(224,226)], [(1,218,160,36),(2,217,161,35),(3,216,162,34),(4,215,163,33),(5,214,164,32),(6,213,165,31),(7,212,166,60),(8,211,167,59),(9,240,168,58),(10,239,169,57),(11,238,170,56),(12,237,171,55),(13,236,172,54),(14,235,173,53),(15,234,174,52),(16,233,175,51),(17,232,176,50),(18,231,177,49),(19,230,178,48),(20,229,179,47),(21,228,180,46),(22,227,151,45),(23,226,152,44),(24,225,153,43),(25,224,154,42),(26,223,155,41),(27,222,156,40),(28,221,157,39),(29,220,158,38),(30,219,159,37),(61,205,109,135),(62,204,110,134),(63,203,111,133),(64,202,112,132),(65,201,113,131),(66,200,114,130),(67,199,115,129),(68,198,116,128),(69,197,117,127),(70,196,118,126),(71,195,119,125),(72,194,120,124),(73,193,91,123),(74,192,92,122),(75,191,93,121),(76,190,94,150),(77,189,95,149),(78,188,96,148),(79,187,97,147),(80,186,98,146),(81,185,99,145),(82,184,100,144),(83,183,101,143),(84,182,102,142),(85,181,103,141),(86,210,104,140),(87,209,105,139),(88,208,106,138),(89,207,107,137),(90,206,108,136)], [(1,69,160,117),(2,88,161,106),(3,77,162,95),(4,66,163,114),(5,85,164,103),(6,74,165,92),(7,63,166,111),(8,82,167,100),(9,71,168,119),(10,90,169,108),(11,79,170,97),(12,68,171,116),(13,87,172,105),(14,76,173,94),(15,65,174,113),(16,84,175,102),(17,73,176,91),(18,62,177,110),(19,81,178,99),(20,70,179,118),(21,89,180,107),(22,78,151,96),(23,67,152,115),(24,86,153,104),(25,75,154,93),(26,64,155,112),(27,83,156,101),(28,72,157,120),(29,61,158,109),(30,80,159,98),(31,192,213,122),(32,181,214,141),(33,200,215,130),(34,189,216,149),(35,208,217,138),(36,197,218,127),(37,186,219,146),(38,205,220,135),(39,194,221,124),(40,183,222,143),(41,202,223,132),(42,191,224,121),(43,210,225,140),(44,199,226,129),(45,188,227,148),(46,207,228,137),(47,196,229,126),(48,185,230,145),(49,204,231,134),(50,193,232,123),(51,182,233,142),(52,201,234,131),(53,190,235,150),(54,209,236,139),(55,198,237,128),(56,187,238,147),(57,206,239,136),(58,195,240,125),(59,184,211,144),(60,203,212,133)])

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222223444444445566610···1012121212121215152020202020···2030···3060···60
size1111303024661010122060222222···2442020202044444412···124···44···4

60 irreducible representations

dim1111111122222222222444444444
type++++++++++-++++++--++-++
imageC1C2C2C2C2C2C2C2S3D4Q8D5D6D6C4○D4D10D10D12C4○D20D42S3S3×Q8S3×D5D4×D5Q8×D5C2×S3×D5D15⋊Q8D5×D12Dic5.D6
kernelD302Q8D304C4C30.Q8C3×C10.D4C5×C4⋊Dic3D303C4C2×D30.C2C2×C15⋊Q8C10.D4C3×Dic5D30C4⋊Dic3C2×Dic5C2×C20C30C2×Dic3C2×C12Dic5C6C10C10C2×C4C6C6C22C2C2C2
# reps1111111112222124248112222444

Matrix representation of D302Q8 in GL4(𝔽61) generated by

1100
60000
00431
00421
,
1100
06000
00044
00430
,
381500
382300
00814
005253
,
60000
06000
004611
004615
G:=sub<GL(4,GF(61))| [1,60,0,0,1,0,0,0,0,0,43,42,0,0,1,1],[1,0,0,0,1,60,0,0,0,0,0,43,0,0,44,0],[38,38,0,0,15,23,0,0,0,0,8,52,0,0,14,53],[60,0,0,0,0,60,0,0,0,0,46,46,0,0,11,15] >;

D302Q8 in GAP, Magma, Sage, TeX

D_{30}\rtimes_2Q_8
% in TeX

G:=Group("D30:2Q8");
// GroupNames label

G:=SmallGroup(480,495);
// by ID

G=gap.SmallGroup(480,495);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,422,135,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^-1,d*a*d^-1=a^19,c*b*c^-1=a^13*b,d*b*d^-1=a^18*b,d*c*d^-1=c^-1>;
// generators/relations

׿
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