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G = D30⋊Q8order 480 = 25·3·5

1st semidirect product of D30 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D301Q8, C6.13(Q8×D5), C51(D63Q8), (C2×C20).16D6, C6.132(D4×D5), C10.13(S3×Q8), C30.38(C2×Q8), Dic3⋊C418D5, (C2×Dic10)⋊6S3, C36(D10⋊Q8), C6.8(C4○D20), C30.132(C2×D4), C1511(C22⋊Q8), (C3×Dic5).8D4, (C6×Dic10)⋊16C2, D303C4.7C2, C30.58(C4○D4), (C2×C12).225D10, C2.15(D15⋊Q8), C30.Q817C2, Dic155C417C2, (C2×Dic5).29D6, D304C4.11C2, (C2×C60).256C22, (C2×C30).101C23, (C2×Dic3).31D10, C2.15(D60⋊C2), C10.13(Q83S3), Dic5.21(C3⋊D4), (C6×Dic5).58C22, (C10×Dic3).61C22, (C2×Dic15).81C22, (C22×D15).32C22, (C2×C4).40(S3×D5), C2.16(D5×C3⋊D4), C10.34(C2×C3⋊D4), C22.170(C2×S3×D5), (C5×Dic3⋊C4)⋊18C2, (C2×D30.C2).4C2, (C2×C6).113(C22×D5), (C2×C10).113(C22×S3), SmallGroup(480,487)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D30⋊Q8
C1C5C15C30C2×C30C6×Dic5C2×D30.C2 — D30⋊Q8
C15C2×C30 — D30⋊Q8
C1C22C2×C4

Generators and relations for D30⋊Q8
 G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a19, cbc-1=a15b, dbd-1=a18b, dcd-1=c-1 >

Subgroups: 796 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, C4×S3 [×2], C2×Dic3 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C3×Q8 [×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, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4 [×2], S3×C2×C4, C6×Q8, C5×Dic3 [×2], C3×Dic5 [×2], C3×Dic5, Dic15, C60, D30 [×2], D30 [×2], C2×C30, C10.D4 [×2], D10⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, D63Q8, D30.C2 [×2], C3×Dic10 [×2], C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15, C2×C60, C22×D15, D10⋊Q8, D304C4, C30.Q8, Dic155C4, C5×Dic3⋊C4, D303C4, C2×D30.C2, C6×Dic10, D30⋊Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D5, D6 [×3], C2×D4, C2×Q8, C4○D4, D10 [×3], C3⋊D4 [×2], C22×S3, C22⋊Q8, C22×D5, S3×Q8, Q83S3, C2×C3⋊D4, S3×D5, C4○D20, D4×D5, Q8×D5, D63Q8, C2×S3×D5, D10⋊Q8, D60⋊C2, D15⋊Q8, D5×C3⋊D4, D30⋊Q8

Smallest permutation representation of D30⋊Q8
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 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(61 86)(62 85)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(87 90)(88 89)(91 97)(92 96)(93 95)(98 120)(99 119)(100 118)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)(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)(151 163)(152 162)(153 161)(154 160)(155 159)(156 158)(164 180)(165 179)(166 178)(167 177)(168 176)(169 175)(170 174)(171 173)(181 190)(182 189)(183 188)(184 187)(185 186)(191 210)(192 209)(193 208)(194 207)(195 206)(196 205)(197 204)(198 203)(199 202)(200 201)(211 218)(212 217)(213 216)(214 215)(219 240)(220 239)(221 238)(222 237)(223 236)(224 235)(225 234)(226 233)(227 232)(228 231)(229 230)
(1 127 89 180)(2 128 90 151)(3 129 61 152)(4 130 62 153)(5 131 63 154)(6 132 64 155)(7 133 65 156)(8 134 66 157)(9 135 67 158)(10 136 68 159)(11 137 69 160)(12 138 70 161)(13 139 71 162)(14 140 72 163)(15 141 73 164)(16 142 74 165)(17 143 75 166)(18 144 76 167)(19 145 77 168)(20 146 78 169)(21 147 79 170)(22 148 80 171)(23 149 81 172)(24 150 82 173)(25 121 83 174)(26 122 84 175)(27 123 85 176)(28 124 86 177)(29 125 87 178)(30 126 88 179)(31 185 101 229)(32 186 102 230)(33 187 103 231)(34 188 104 232)(35 189 105 233)(36 190 106 234)(37 191 107 235)(38 192 108 236)(39 193 109 237)(40 194 110 238)(41 195 111 239)(42 196 112 240)(43 197 113 211)(44 198 114 212)(45 199 115 213)(46 200 116 214)(47 201 117 215)(48 202 118 216)(49 203 119 217)(50 204 120 218)(51 205 91 219)(52 206 92 220)(53 207 93 221)(54 208 94 222)(55 209 95 223)(56 210 96 224)(57 181 97 225)(58 182 98 226)(59 183 99 227)(60 184 100 228)
(1 230 89 186)(2 219 90 205)(3 238 61 194)(4 227 62 183)(5 216 63 202)(6 235 64 191)(7 224 65 210)(8 213 66 199)(9 232 67 188)(10 221 68 207)(11 240 69 196)(12 229 70 185)(13 218 71 204)(14 237 72 193)(15 226 73 182)(16 215 74 201)(17 234 75 190)(18 223 76 209)(19 212 77 198)(20 231 78 187)(21 220 79 206)(22 239 80 195)(23 228 81 184)(24 217 82 203)(25 236 83 192)(26 225 84 181)(27 214 85 200)(28 233 86 189)(29 222 87 208)(30 211 88 197)(31 138 101 161)(32 127 102 180)(33 146 103 169)(34 135 104 158)(35 124 105 177)(36 143 106 166)(37 132 107 155)(38 121 108 174)(39 140 109 163)(40 129 110 152)(41 148 111 171)(42 137 112 160)(43 126 113 179)(44 145 114 168)(45 134 115 157)(46 123 116 176)(47 142 117 165)(48 131 118 154)(49 150 119 173)(50 139 120 162)(51 128 91 151)(52 147 92 170)(53 136 93 159)(54 125 94 178)(55 144 95 167)(56 133 96 156)(57 122 97 175)(58 141 98 164)(59 130 99 153)(60 149 100 172)

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

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

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

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○D4D10D10C3⋊D4C4○D20S3×Q8Q83S3S3×D5D4×D5Q8×D5C2×S3×D5D60⋊C2D15⋊Q8D5×C3⋊D4
kernelD30⋊Q8D304C4C30.Q8Dic155C4C5×Dic3⋊C4D303C4C2×D30.C2C6×Dic10C2×Dic10C3×Dic5D30Dic3⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12Dic5C6C10C10C2×C4C6C6C22C2C2C2
# reps1111111112222124248112222444

Matrix representation of D30⋊Q8 in GL6(𝔽61)

1200000
9590000
00431900
0060100
0000600
0000060
,
1200000
0600000
0004300
0044000
0000600
0000221
,
100000
010000
00251100
00543600
0000555
0000546
,
6000000
0600000
00153500
0044600
000010
000001

G:=sub<GL(6,GF(61))| [1,9,0,0,0,0,20,59,0,0,0,0,0,0,43,60,0,0,0,0,19,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,20,60,0,0,0,0,0,0,0,44,0,0,0,0,43,0,0,0,0,0,0,0,60,22,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,25,54,0,0,0,0,11,36,0,0,0,0,0,0,55,54,0,0,0,0,5,6],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,15,4,0,0,0,0,35,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D30⋊Q8 in GAP, Magma, Sage, TeX

D_{30}\rtimes Q_8
% in TeX

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

G:=SmallGroup(480,487);
// by ID

G=gap.SmallGroup(480,487);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,254,219,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^19,c*b*c^-1=a^15*b,d*b*d^-1=a^18*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽