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G = Dic10.D6order 480 = 25·3·5

1st non-split extension by Dic10 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D151Q16, C40.17D6, Dic124D5, Dic204S3, D30.22D4, C24.17D10, Dic10.1D6, Dic6.1D10, C60.143C23, C120.36C22, Dic15.27D4, C52(S3×Q16), C32(D5×Q16), C153(C2×Q16), C8.29(S3×D5), C6.32(D4×D5), (C8×D15).2C2, C15⋊Q169C2, C10.32(S3×D4), C30.18(C2×D4), D15⋊Q8.2C2, (C5×Dic12)⋊6C2, (C3×Dic20)⋊7C2, C20.75(C22×S3), C12.74(C22×D5), C2.10(C20⋊D6), C153C8.40C22, (C4×D15).55C22, (C5×Dic6).27C22, (C3×Dic10).27C22, C4.116(C2×S3×D5), SmallGroup(480,340)

Series: Derived Chief Lower central Upper central

C1C60 — Dic10.D6
C1C5C15C30C60C3×Dic10D15⋊Q8 — Dic10.D6
C15C30C60 — Dic10.D6
C1C2C4C8

Generators and relations for Dic10.D6
 G = < a,b,c,d | a12=d2=1, b2=c10=a6, bab-1=cac-1=a-1, dad=a5, cbc-1=a3b, bd=db, dcd=c9 >

Subgroups: 636 in 120 conjugacy classes, 40 normal (26 characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×5], C22, C5, S3 [×2], C6, C8, C8, C2×C4 [×3], Q8 [×6], D5 [×2], C10, Dic3 [×3], C12, C12 [×2], D6, C15, C2×C8, Q16 [×4], C2×Q8 [×2], Dic5 [×3], C20, C20 [×2], D10, C3⋊C8, C24, Dic6 [×2], Dic6 [×2], C4×S3 [×3], C3×Q8 [×2], D15 [×2], C30, C2×Q16, C52C8, C40, Dic10 [×2], Dic10 [×2], C4×D5 [×3], C5×Q8 [×2], S3×C8, Dic12, C3⋊Q16 [×2], C3×Q16, S3×Q8 [×2], C5×Dic3 [×2], C3×Dic5 [×2], Dic15, C60, D30, C8×D5, Dic20, C5⋊Q16 [×2], C5×Q16, Q8×D5 [×2], S3×Q16, C153C8, C120, D30.C2 [×2], C15⋊Q8 [×2], C3×Dic10 [×2], C5×Dic6 [×2], C4×D15, D5×Q16, C15⋊Q16 [×2], C3×Dic20, C5×Dic12, C8×D15, D15⋊Q8 [×2], Dic10.D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], Q16 [×2], C2×D4, D10 [×3], C22×S3, C2×Q16, C22×D5, S3×D4, S3×D5, D4×D5, S3×Q16, C2×S3×D5, D5×Q16, C20⋊D6, Dic10.D6

Smallest permutation representation of Dic10.D6
On 240 points
Generators in S240
(1 225 79 144 123 218 11 235 69 154 133 208)(2 209 134 155 70 236 12 219 124 145 80 226)(3 227 61 146 125 220 13 237 71 156 135 210)(4 211 136 157 72 238 14 201 126 147 62 228)(5 229 63 148 127 202 15 239 73 158 137 212)(6 213 138 159 74 240 16 203 128 149 64 230)(7 231 65 150 129 204 17 221 75 160 139 214)(8 215 140 141 76 222 18 205 130 151 66 232)(9 233 67 152 131 206 19 223 77 142 121 216)(10 217 122 143 78 224 20 207 132 153 68 234)(21 91 162 114 198 41 31 81 172 104 188 51)(22 52 189 105 173 82 32 42 199 115 163 92)(23 93 164 116 200 43 33 83 174 106 190 53)(24 54 191 107 175 84 34 44 181 117 165 94)(25 95 166 118 182 45 35 85 176 108 192 55)(26 56 193 109 177 86 36 46 183 119 167 96)(27 97 168 120 184 47 37 87 178 110 194 57)(28 58 195 111 179 88 38 48 185 101 169 98)(29 99 170 102 186 49 39 89 180 112 196 59)(30 60 197 113 161 90 40 50 187 103 171 100)
(1 161 11 171)(2 41 12 51)(3 163 13 173)(4 43 14 53)(5 165 15 175)(6 45 16 55)(7 167 17 177)(8 47 18 57)(9 169 19 179)(10 49 20 59)(21 226 31 236)(22 125 32 135)(23 228 33 238)(24 127 34 137)(25 230 35 240)(26 129 36 139)(27 232 37 222)(28 131 38 121)(29 234 39 224)(30 133 40 123)(42 156 52 146)(44 158 54 148)(46 160 56 150)(48 142 58 152)(50 144 60 154)(61 199 71 189)(62 83 72 93)(63 181 73 191)(64 85 74 95)(65 183 75 193)(66 87 76 97)(67 185 77 195)(68 89 78 99)(69 187 79 197)(70 91 80 81)(82 210 92 220)(84 212 94 202)(86 214 96 204)(88 216 98 206)(90 218 100 208)(101 223 111 233)(102 132 112 122)(103 225 113 235)(104 134 114 124)(105 227 115 237)(106 136 116 126)(107 229 117 239)(108 138 118 128)(109 231 119 221)(110 140 120 130)(141 168 151 178)(143 170 153 180)(145 172 155 162)(147 174 157 164)(149 176 159 166)(182 203 192 213)(184 205 194 215)(186 207 196 217)(188 209 198 219)(190 211 200 201)
(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 15)(2 4)(3 13)(5 11)(6 20)(7 9)(8 18)(10 16)(12 14)(17 19)(21 200)(22 189)(23 198)(24 187)(25 196)(26 185)(27 194)(28 183)(29 192)(30 181)(31 190)(32 199)(33 188)(34 197)(35 186)(36 195)(37 184)(38 193)(39 182)(40 191)(41 43)(42 52)(44 50)(45 59)(46 48)(47 57)(49 55)(51 53)(54 60)(56 58)(61 125)(62 134)(63 123)(64 132)(65 121)(66 130)(67 139)(68 128)(69 137)(70 126)(71 135)(72 124)(73 133)(74 122)(75 131)(76 140)(77 129)(78 138)(79 127)(80 136)(81 116)(82 105)(83 114)(84 103)(85 112)(86 101)(87 110)(88 119)(89 108)(90 117)(91 106)(92 115)(93 104)(94 113)(95 102)(96 111)(97 120)(98 109)(99 118)(100 107)(141 151)(142 160)(143 149)(144 158)(145 147)(146 156)(148 154)(150 152)(153 159)(155 157)(161 175)(162 164)(163 173)(165 171)(166 180)(167 169)(168 178)(170 176)(172 174)(177 179)(201 226)(202 235)(203 224)(204 233)(205 222)(206 231)(207 240)(208 229)(209 238)(210 227)(211 236)(212 225)(213 234)(214 223)(215 232)(216 221)(217 230)(218 239)(219 228)(220 237)

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

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

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

51 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B 6 8A8B8C8D10A10B12A12B12C15A15B20A20B20C20D20E20F24A24B30A30B40A40B40C40D60A60B60C60D120A···120H
order12223444444556888810101212121515202020202020242430304040404060606060120···120
size11151522121220203022222303022440404444242424244444444444444···4

51 irreducible representations

dim11111122222222244444444
type++++++++++++-+++++-+-
imageC1C2C2C2C2C2S3D4D4D5D6D6Q16D10D10S3×D4S3×D5D4×D5S3×Q16C2×S3×D5D5×Q16C20⋊D6Dic10.D6
kernelDic10.D6C15⋊Q16C3×Dic20C5×Dic12C8×D15D15⋊Q8Dic20Dic15D30Dic12C40Dic10D15C24Dic6C10C8C6C5C4C3C2C1
# reps12111211121242412222448

Matrix representation of Dic10.D6 in GL6(𝔽241)

010000
24000000
00240100
00240000
000010
000001
,
2091360000
136320000
00024000
00240000
00002400
00000240
,
61800000
801800000
000100
001000
00000190
000052189
,
24000000
02400000
000100
001000
000052190
000053189

G:=sub<GL(6,GF(241))| [0,240,0,0,0,0,1,0,0,0,0,0,0,0,240,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[209,136,0,0,0,0,136,32,0,0,0,0,0,0,0,240,0,0,0,0,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[61,80,0,0,0,0,80,180,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,52,0,0,0,0,190,189],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,52,53,0,0,0,0,190,189] >;

Dic10.D6 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}.D_6
% in TeX

G:=Group("Dic10.D6");
// GroupNames label

G:=SmallGroup(480,340);
// by ID

G=gap.SmallGroup(480,340);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,135,142,675,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^12=d^2=1,b^2=c^10=a^6,b*a*b^-1=c*a*c^-1=a^-1,d*a*d=a^5,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=c^9>;
// generators/relations

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