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G = D3010Q8order 480 = 25·3·5

6th semidirect product of D30 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D3010Q8, C60.92D4, C12.22D20, C6.11(Q8×D5), C4⋊Dic312D5, C52(D63Q8), C6.59(C2×D20), C159(C22⋊Q8), C10.11(S3×Q8), C30.33(C2×Q8), C34(D102Q8), C30.123(C2×D4), (C2×C20).122D6, (C2×Dic10)⋊11S3, (C6×Dic10)⋊11C2, D304C4.8C2, C30.51(C4○D4), (C2×C12).123D10, C20.38(C3⋊D4), C4.25(C3⋊D20), C2.14(D15⋊Q8), (C2×C30).80C23, C6.Dic1010C2, (C2×Dic5).28D6, C6.10(D42D5), (C2×C60).199C22, (C2×Dic3).28D10, C2.14(D12⋊D5), C10.29(Q83S3), (C6×Dic5).47C22, (C10×Dic3).47C22, (C2×Dic15).197C22, (C22×D15).100C22, (C5×C4⋊Dic3)⋊9C2, (C2×C4×D15).15C2, (C2×C4).209(S3×D5), C2.18(C2×C3⋊D20), C10.14(C2×C3⋊D4), C22.165(C2×S3×D5), (C2×C6).92(C22×D5), (C2×C10).92(C22×S3), SmallGroup(480,466)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D3010Q8
C1C5C15C30C2×C30C6×Dic5C6.Dic10 — D3010Q8
C15C2×C30 — D3010Q8
C1C22C2×C4

Generators and relations for D3010Q8
 G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a19, bc=cb, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 796 in 148 conjugacy classes, 54 normal (32 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×7], Q8 [×2], C23, D5 [×2], C10 [×3], Dic3 [×3], C12 [×2], C12 [×2], D6 [×4], C2×C6, C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], 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 [×2], C4⋊Dic3, D6⋊C4 [×2], S3×C2×C4, C6×Q8, C5×Dic3 [×2], C3×Dic5 [×2], Dic15, C60 [×2], D30 [×2], D30 [×2], C2×C30, C4⋊Dic5 [×2], D10⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, D63Q8, C3×Dic10 [×2], C6×Dic5 [×2], C10×Dic3 [×2], C4×D15 [×2], C2×Dic15, C2×C60, C22×D15, D102Q8, D304C4 [×2], C6.Dic10 [×2], C5×C4⋊Dic3, C6×Dic10, C2×C4×D15, D3010Q8
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, D20 [×2], C22×D5, S3×Q8, Q83S3, C2×C3⋊D4, S3×D5, C2×D20, D42D5, Q8×D5, D63Q8, C3⋊D20 [×2], C2×S3×D5, D102Q8, D12⋊D5, D15⋊Q8, C2×C3⋊D20, D3010Q8

Smallest permutation representation of D3010Q8
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)
(1 230)(2 229)(3 228)(4 227)(5 226)(6 225)(7 224)(8 223)(9 222)(10 221)(11 220)(12 219)(13 218)(14 217)(15 216)(16 215)(17 214)(18 213)(19 212)(20 211)(21 240)(22 239)(23 238)(24 237)(25 236)(26 235)(27 234)(28 233)(29 232)(30 231)(31 128)(32 127)(33 126)(34 125)(35 124)(36 123)(37 122)(38 121)(39 150)(40 149)(41 148)(42 147)(43 146)(44 145)(45 144)(46 143)(47 142)(48 141)(49 140)(50 139)(51 138)(52 137)(53 136)(54 135)(55 134)(56 133)(57 132)(58 131)(59 130)(60 129)(61 185)(62 184)(63 183)(64 182)(65 181)(66 210)(67 209)(68 208)(69 207)(70 206)(71 205)(72 204)(73 203)(74 202)(75 201)(76 200)(77 199)(78 198)(79 197)(80 196)(81 195)(82 194)(83 193)(84 192)(85 191)(86 190)(87 189)(88 188)(89 187)(90 186)(91 173)(92 172)(93 171)(94 170)(95 169)(96 168)(97 167)(98 166)(99 165)(100 164)(101 163)(102 162)(103 161)(104 160)(105 159)(106 158)(107 157)(108 156)(109 155)(110 154)(111 153)(112 152)(113 151)(114 180)(115 179)(116 178)(117 177)(118 176)(119 175)(120 174)
(1 100 231 165)(2 101 232 166)(3 102 233 167)(4 103 234 168)(5 104 235 169)(6 105 236 170)(7 106 237 171)(8 107 238 172)(9 108 239 173)(10 109 240 174)(11 110 211 175)(12 111 212 176)(13 112 213 177)(14 113 214 178)(15 114 215 179)(16 115 216 180)(17 116 217 151)(18 117 218 152)(19 118 219 153)(20 119 220 154)(21 120 221 155)(22 91 222 156)(23 92 223 157)(24 93 224 158)(25 94 225 159)(26 95 226 160)(27 96 227 161)(28 97 228 162)(29 98 229 163)(30 99 230 164)(31 81 122 189)(32 82 123 190)(33 83 124 191)(34 84 125 192)(35 85 126 193)(36 86 127 194)(37 87 128 195)(38 88 129 196)(39 89 130 197)(40 90 131 198)(41 61 132 199)(42 62 133 200)(43 63 134 201)(44 64 135 202)(45 65 136 203)(46 66 137 204)(47 67 138 205)(48 68 139 206)(49 69 140 207)(50 70 141 208)(51 71 142 209)(52 72 143 210)(53 73 144 181)(54 74 145 182)(55 75 146 183)(56 76 147 184)(57 77 148 185)(58 78 149 186)(59 79 150 187)(60 80 121 188)
(1 185 231 77)(2 204 232 66)(3 193 233 85)(4 182 234 74)(5 201 235 63)(6 190 236 82)(7 209 237 71)(8 198 238 90)(9 187 239 79)(10 206 240 68)(11 195 211 87)(12 184 212 76)(13 203 213 65)(14 192 214 84)(15 181 215 73)(16 200 216 62)(17 189 217 81)(18 208 218 70)(19 197 219 89)(20 186 220 78)(21 205 221 67)(22 194 222 86)(23 183 223 75)(24 202 224 64)(25 191 225 83)(26 210 226 72)(27 199 227 61)(28 188 228 80)(29 207 229 69)(30 196 230 88)(31 116 122 151)(32 105 123 170)(33 94 124 159)(34 113 125 178)(35 102 126 167)(36 91 127 156)(37 110 128 175)(38 99 129 164)(39 118 130 153)(40 107 131 172)(41 96 132 161)(42 115 133 180)(43 104 134 169)(44 93 135 158)(45 112 136 177)(46 101 137 166)(47 120 138 155)(48 109 139 174)(49 98 140 163)(50 117 141 152)(51 106 142 171)(52 95 143 160)(53 114 144 179)(54 103 145 168)(55 92 146 157)(56 111 147 176)(57 100 148 165)(58 119 149 154)(59 108 150 173)(60 97 121 162)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222223444444445566610···1012121212121215152020202020···2030···3060···60
size11113030222121220203030222222···2442020202044444412···124···44···4

60 irreducible representations

dim11111122222222222444444444
type++++++++-++++++-++--++
imageC1C2C2C2C2C2S3D4Q8D5D6D6C4○D4D10D10C3⋊D4D20S3×Q8Q83S3S3×D5D42D5Q8×D5C3⋊D20C2×S3×D5D12⋊D5D15⋊Q8
kernelD3010Q8D304C4C6.Dic10C5×C4⋊Dic3C6×Dic10C2×C4×D15C2×Dic10C60D30C4⋊Dic3C2×Dic5C2×C20C30C2×Dic3C2×C12C20C12C10C10C2×C4C6C6C4C22C2C2
# reps12211112222124248112224244

Matrix representation of D3010Q8 in GL6(𝔽61)

1810000
6000000
000100
00606000
0000600
0000060
,
23530000
5380000
000100
001000
000010
00005660
,
6000000
0600000
001000
000100
0000500
00005511
,
47590000
6140000
001000
000100
00002358
00001438

G:=sub<GL(6,GF(61))| [18,60,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[23,5,0,0,0,0,53,38,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,56,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,50,55,0,0,0,0,0,11],[47,6,0,0,0,0,59,14,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,23,14,0,0,0,0,58,38] >;

D3010Q8 in GAP, Magma, Sage, TeX

D_{30}\rtimes_{10}Q_8
% in TeX

G:=Group("D30:10Q8");
// GroupNames label

G:=SmallGroup(480,466);
// by ID

G=gap.SmallGroup(480,466);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,64,422,219,100,1356,18822]);
// 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,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽