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G = D10.17D12order 480 = 25·3·5

6th non-split extension by D10 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.17D12, D6⋊C44D5, (C6×D5).7D4, C605C44C2, C6.15(D4×D5), D6⋊Dic57C2, C2.18(D5×D12), C30.39(C2×D4), (C2×C20).17D6, D10⋊C44S3, C10.15(C2×D12), (C2×C12).13D10, (C2×C60).6C22, C6.54(C4○D20), C30.Q820C2, (C2×Dic5).31D6, (C22×D5).48D6, C30.117(C4○D4), C6.70(D42D5), C33(D10.12D4), (C2×C30).104C23, (C22×S3).11D10, C52(C23.21D6), C158(C22.D4), C10.28(D42S3), C2.15(D205S3), (C2×Dic3).101D10, (C6×Dic5).60C22, C2.15(C30.C23), (C2×Dic15).84C22, (C10×Dic3).64C22, (C5×D6⋊C4)⋊4C2, (C2×D5×Dic3)⋊7C2, (C2×C4).42(S3×D5), (C2×C15⋊D4).1C2, (C3×D10⋊C4)⋊4C2, (D5×C2×C6).18C22, C22.172(C2×S3×D5), (S3×C2×C10).16C22, (C2×C6).116(C22×D5), (C2×C10).116(C22×S3), SmallGroup(480,490)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D10.17D12
C1C5C15C30C2×C30D5×C2×C6C2×D5×Dic3 — D10.17D12
C15C2×C30 — D10.17D12
C1C22C2×C4

Generators and relations for D10.17D12
 G = < a,b,c,d | a10=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a5b, dcd=a5c-1 >

Subgroups: 812 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 [×2], C10 [×3], C10, Dic3 [×3], C12 [×2], D6 [×3], C2×C6, C2×C6 [×4], C15, C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, Dic5 [×3], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×3], C2×Dic3, C2×Dic3 [×4], C3⋊D4 [×2], C2×C12, C2×C12, C22×S3, C22×C6, C5×S3, C3×D5 [×2], C30 [×3], C22.D4, C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C2×C20, C22×D5, C22×C10, C4⋊Dic3 [×2], D6⋊C4, D6⋊C4, C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15 [×2], C60, C6×D5 [×2], C6×D5 [×2], S3×C10 [×3], C2×C30, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C2×C5⋊D4, C23.21D6, D5×Dic3 [×2], C15⋊D4 [×2], C6×Dic5, C10×Dic3, C2×Dic15 [×2], C2×C60, D5×C2×C6, S3×C2×C10, D10.12D4, D6⋊Dic5, C30.Q8, C3×D10⋊C4, C5×D6⋊C4, C605C4, C2×D5×Dic3, C2×C15⋊D4, D10.17D12
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, D42D5, C23.21D6, C2×S3×D5, D10.12D4, D205S3, D5×D12, C30.C23, D10.17D12

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

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G5A5B6A6B6C6D6E10A···10F10G10H10I10J12A12B12C12D15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order122222234444444556666610···1010101010121212121515202020202020202030···3060···60
size11111010122466203030602222220202···212121212442020444444121212124···44···4

60 irreducible representations

dim1111111122222222222244444444
type++++++++++++++++++-++-+-+-
imageC1C2C2C2C2C2C2C2S3D4D5D6D6D6C4○D4D10D10D10D12C4○D20D42S3S3×D5D4×D5D42D5C2×S3×D5D205S3D5×D12C30.C23
kernelD10.17D12D6⋊Dic5C30.Q8C3×D10⋊C4C5×D6⋊C4C605C4C2×D5×Dic3C2×C15⋊D4D10⋊C4C6×D5D6⋊C4C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12C22×S3D10C6C10C2×C4C6C6C22C2C2C2
# reps1111111112211142224822222444

Matrix representation of D10.17D12 in GL8(𝔽61)

600000000
060000000
0044440000
0017600000
00001000
00000100
00000010
00000001
,
600000000
21000000
0044440000
0060170000
000060000
000006000
00000010
00000001
,
6060000000
21000000
00100000
00010000
00000100
000060000
000000060
000000160
,
5050000000
2211000000
00100000
00010000
0000475400
0000541400
0000003012
0000004231

G:=sub<GL(8,GF(61))| [60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,44,17,0,0,0,0,0,0,44,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[60,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,44,60,0,0,0,0,0,0,44,17,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[60,2,0,0,0,0,0,0,60,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60],[50,22,0,0,0,0,0,0,50,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,47,54,0,0,0,0,0,0,54,14,0,0,0,0,0,0,0,0,30,42,0,0,0,0,0,0,12,31] >;

D10.17D12 in GAP, Magma, Sage, TeX

D_{10}._{17}D_{12}
% in TeX

G:=Group("D10.17D12");
// GroupNames label

G:=SmallGroup(480,490);
// by ID

G=gap.SmallGroup(480,490);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,254,219,142,1356,18822]);
// Polycyclic

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

׿
×
𝔽