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G = D10.16D12order 480 = 25·3·5

5th non-split extension by D10 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.16D12, C4⋊Dic36D5, (C6×D5).6D4, C6.14(D4×D5), C2.17(D5×D12), C30.38(C2×D4), C10.14(C2×D12), (C2×C12).12D10, (C2×C20).223D6, D10⋊C416S3, D303C419C2, D304C412C2, C30.59(C4○D4), C6.71(C4○D20), C30.Q819C2, (C2×Dic5).30D6, (C22×D5).47D6, C34(D10.13D4), (C2×C60).316C22, (C2×C30).103C23, C6.34(Q82D5), (C2×Dic3).32D10, C157(C22.D4), C51(C23.21D6), C10.45(D42S3), C2.16(D20⋊S3), (C6×Dic5).59C22, C2.17(Dic5.D6), (C10×Dic3).63C22, (C2×Dic15).83C22, (C22×D15).33C22, (C2×D5×Dic3)⋊6C2, (C2×C4).41(S3×D5), (C5×C4⋊Dic3)⋊17C2, (C2×C3⋊D20).7C2, (D5×C2×C6).17C22, C22.171(C2×S3×D5), (C3×D10⋊C4)⋊21C2, (C2×C6).115(C22×D5), (C2×C10).115(C22×S3), SmallGroup(480,489)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D10.16D12
C1C5C15C30C2×C30D5×C2×C6C2×D5×Dic3 — D10.16D12
C15C2×C30 — D10.16D12
C1C22C2×C4

Generators and relations for D10.16D12
 G = < a,b,c,d | a10=b2=c12=1, d2=a5, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a5b, dcd-1=c-1 >

Subgroups: 940 in 156 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×5], C22, C22 [×7], C5, S3, C6 [×3], C6 [×2], C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], D5 [×3], C10 [×3], Dic3 [×3], C12 [×2], D6 [×3], C2×C6, C2×C6 [×4], C15, C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×2], C20 [×3], D10 [×2], D10 [×5], C2×C10, C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×2], C2×C12, C2×C12, C22×S3, C22×C6, C3×D5 [×2], D15, C30 [×3], C22.D4, C4×D5 [×2], D20 [×2], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5, C4⋊Dic3, C4⋊Dic3, D6⋊C4 [×2], C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C5×Dic3 [×2], C3×Dic5, Dic15, C60, C6×D5 [×2], C6×D5 [×2], D30 [×3], C2×C30, C10.D4, D10⋊C4, D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×D20, C23.21D6, D5×Dic3 [×2], C3⋊D20 [×2], C6×Dic5, C10×Dic3 [×2], C2×Dic15, C2×C60, D5×C2×C6, C22×D15, D10.13D4, D304C4, C30.Q8, C3×D10⋊C4, C5×C4⋊Dic3, D303C4, C2×D5×Dic3, C2×C3⋊D20, D10.16D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], D12 [×2], C22×S3, C22.D4, C22×D5, C2×D12, D42S3 [×2], S3×D5, C4○D20, D4×D5, Q82D5, C23.21D6, C2×S3×D5, D10.13D4, D20⋊S3, D5×D12, Dic5.D6, D10.16D12

Smallest permutation representation of D10.16D12
On 240 points
Generators in S240
(1 116 107 230 38 164 221 26 178 212)(2 117 108 231 39 165 222 27 179 213)(3 118 97 232 40 166 223 28 180 214)(4 119 98 233 41 167 224 29 169 215)(5 120 99 234 42 168 225 30 170 216)(6 109 100 235 43 157 226 31 171 205)(7 110 101 236 44 158 227 32 172 206)(8 111 102 237 45 159 228 33 173 207)(9 112 103 238 46 160 217 34 174 208)(10 113 104 239 47 161 218 35 175 209)(11 114 105 240 48 162 219 36 176 210)(12 115 106 229 37 163 220 25 177 211)(13 53 200 144 123 183 153 66 78 86)(14 54 201 133 124 184 154 67 79 87)(15 55 202 134 125 185 155 68 80 88)(16 56 203 135 126 186 156 69 81 89)(17 57 204 136 127 187 145 70 82 90)(18 58 193 137 128 188 146 71 83 91)(19 59 194 138 129 189 147 72 84 92)(20 60 195 139 130 190 148 61 73 93)(21 49 196 140 131 191 149 62 74 94)(22 50 197 141 132 192 150 63 75 95)(23 51 198 142 121 181 151 64 76 96)(24 52 199 143 122 182 152 65 77 85)
(1 206)(2 45)(3 208)(4 47)(5 210)(6 37)(7 212)(8 39)(9 214)(10 41)(11 216)(12 43)(13 84)(14 139)(15 74)(16 141)(17 76)(18 143)(19 78)(20 133)(21 80)(22 135)(23 82)(24 137)(25 31)(26 101)(27 33)(28 103)(29 35)(30 105)(32 107)(34 97)(36 99)(38 158)(40 160)(42 162)(44 164)(46 166)(48 168)(49 68)(50 203)(51 70)(52 193)(53 72)(54 195)(55 62)(56 197)(57 64)(58 199)(59 66)(60 201)(61 154)(63 156)(65 146)(67 148)(69 150)(71 152)(73 184)(75 186)(77 188)(79 190)(81 192)(83 182)(85 128)(86 92)(87 130)(88 94)(89 132)(90 96)(91 122)(93 124)(95 126)(98 104)(100 106)(102 108)(109 229)(110 178)(111 231)(112 180)(113 233)(114 170)(115 235)(116 172)(117 237)(118 174)(119 239)(120 176)(121 127)(123 129)(125 131)(134 191)(136 181)(138 183)(140 185)(142 187)(144 189)(145 198)(147 200)(149 202)(151 204)(153 194)(155 196)(157 211)(159 213)(161 215)(163 205)(165 207)(167 209)(169 218)(171 220)(173 222)(175 224)(177 226)(179 228)(217 232)(219 234)(221 236)(223 238)(225 240)(227 230)
(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 70 164 204)(2 69 165 203)(3 68 166 202)(4 67 167 201)(5 66 168 200)(6 65 157 199)(7 64 158 198)(8 63 159 197)(9 62 160 196)(10 61 161 195)(11 72 162 194)(12 71 163 193)(13 170 183 234)(14 169 184 233)(15 180 185 232)(16 179 186 231)(17 178 187 230)(18 177 188 229)(19 176 189 240)(20 175 190 239)(21 174 191 238)(22 173 192 237)(23 172 181 236)(24 171 182 235)(25 128 106 91)(26 127 107 90)(27 126 108 89)(28 125 97 88)(29 124 98 87)(30 123 99 86)(31 122 100 85)(32 121 101 96)(33 132 102 95)(34 131 103 94)(35 130 104 93)(36 129 105 92)(37 58 211 146)(38 57 212 145)(39 56 213 156)(40 55 214 155)(41 54 215 154)(42 53 216 153)(43 52 205 152)(44 51 206 151)(45 50 207 150)(46 49 208 149)(47 60 209 148)(48 59 210 147)(73 218 139 113)(74 217 140 112)(75 228 141 111)(76 227 142 110)(77 226 143 109)(78 225 144 120)(79 224 133 119)(80 223 134 118)(81 222 135 117)(82 221 136 116)(83 220 137 115)(84 219 138 114)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A6B6C6D6E10A···10F12A12B12C12D15A15B20A20B20C20D20E···20L30A···30F60A···60H
order122222234444444556666610···101212121215152020202020···2030···3060···60
size11111010602466122030302222220202···244202044444412···124···44···4

60 irreducible representations

dim111111112222222222244444444
type+++++++++++++++++-+++++
imageC1C2C2C2C2C2C2C2S3D4D5D6D6D6C4○D4D10D10D12C4○D20D42S3S3×D5D4×D5Q82D5C2×S3×D5D20⋊S3D5×D12Dic5.D6
kernelD10.16D12D304C4C30.Q8C3×D10⋊C4C5×C4⋊Dic3D303C4C2×D5×Dic3C2×C3⋊D20D10⋊C4C6×D5C4⋊Dic3C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12D10C6C10C2×C4C6C6C22C2C2C2
# reps111111111221114424822222444

Matrix representation of D10.16D12 in GL4(𝔽61) generated by

181700
43000
0010
0001
,
436000
181800
00600
00060
,
301600
13100
005333
00240
,
255400
113600
001912
003142
G:=sub<GL(4,GF(61))| [18,43,0,0,17,0,0,0,0,0,1,0,0,0,0,1],[43,18,0,0,60,18,0,0,0,0,60,0,0,0,0,60],[30,1,0,0,16,31,0,0,0,0,53,24,0,0,33,0],[25,11,0,0,54,36,0,0,0,0,19,31,0,0,12,42] >;

D10.16D12 in GAP, Magma, Sage, TeX

D_{10}._{16}D_{12}
% in TeX

G:=Group("D10.16D12");
// GroupNames label

G:=SmallGroup(480,489);
// by ID

G=gap.SmallGroup(480,489);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,254,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^12=1,d^2=a^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽