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G = D10.Dic6order 480 = 25·3·5

1st non-split extension by D10 of Dic6 acting via Dic6/Dic3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.1Dic6, C3⋊C8.1F5, C12.7(C2×F5), C30.3(C4⋊C4), C4.17(S3×F5), (C6×D5).1Q8, C153C8.1C4, C20.17(C4×S3), C4.F5.1S3, C60.17(C2×C4), (C4×D5).62D6, C6.10(C4⋊F5), C32(C40.C4), C151(C8.C4), C12.F5.1C2, C51(C12.53D4), C2.6(Dic3⋊F5), (C3×Dic5).28D4, C10.3(Dic3⋊C4), (D5×C12).48C22, Dic5.16(C3⋊D4), (C5×C3⋊C8).1C4, (D5×C3⋊C8).4C2, (C3×C4.F5).1C2, SmallGroup(480,237)

Series: Derived Chief Lower central Upper central

C1C60 — D10.Dic6
C1C5C15C30C3×Dic5D5×C12C3×C4.F5 — D10.Dic6
C15C30C60 — D10.Dic6
C1C2C4

Generators and relations for D10.Dic6
 G = < a,b,c,d | a10=b2=1, c12=a5, d2=a-1bc6, bab=a-1, cac-1=a3, ad=da, cbc-1=a7b, bd=db, dcd-1=a-1bc11 >

Subgroups: 260 in 60 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8 [×4], C2×C4, D5, C10, C12, C12, C2×C6, C15, C2×C8, M4(2) [×2], Dic5, C20, D10, C3⋊C8, C3⋊C8 [×2], C24, C2×C12, C3×D5, C30, C8.C4, C52C8, C40, C5⋊C8 [×2], C4×D5, C2×C3⋊C8, C4.Dic3, C3×M4(2), C3×Dic5, C60, C6×D5, C8×D5, C4.F5, C4.F5, C12.53D4, C5×C3⋊C8, C153C8, C3×C5⋊C8, C15⋊C8, D5×C12, C40.C4, D5×C3⋊C8, C3×C4.F5, C12.F5, D10.Dic6
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, F5, Dic6, C4×S3, C3⋊D4, C8.C4, C2×F5, Dic3⋊C4, C4⋊F5, C12.53D4, S3×F5, C40.C4, Dic3⋊F5, D10.Dic6

Smallest permutation representation of D10.Dic6
On 240 points
Generators in S240
(1 234 140 155 186 13 222 128 167 174)(2 156 223 175 141 14 168 235 187 129)(3 176 145 130 224 15 188 157 142 236)(4 131 189 237 146 16 143 177 225 158)(5 238 144 159 190 17 226 132 147 178)(6 160 227 179 121 18 148 239 191 133)(7 180 149 134 228 19 192 161 122 240)(8 135 169 217 150 20 123 181 229 162)(9 218 124 163 170 21 230 136 151 182)(10 164 231 183 125 22 152 219 171 137)(11 184 153 138 232 23 172 165 126 220)(12 139 173 221 154 24 127 185 233 166)(25 75 105 58 211 37 87 117 70 199)(26 59 88 200 106 38 71 76 212 118)(27 201 72 119 89 39 213 60 107 77)(28 120 214 78 49 40 108 202 90 61)(29 79 109 62 215 41 91 97 50 203)(30 63 92 204 110 42 51 80 216 98)(31 205 52 99 93 43 193 64 111 81)(32 100 194 82 53 44 112 206 94 65)(33 83 113 66 195 45 95 101 54 207)(34 67 96 208 114 46 55 84 196 102)(35 209 56 103 73 47 197 68 115 85)(36 104 198 86 57 48 116 210 74 69)
(1 174)(2 141)(3 236)(4 146)(5 178)(6 121)(7 240)(8 150)(9 182)(10 125)(11 220)(12 154)(13 186)(14 129)(15 224)(16 158)(17 190)(18 133)(19 228)(20 162)(21 170)(22 137)(23 232)(24 166)(25 75)(26 212)(27 60)(29 79)(30 216)(31 64)(33 83)(34 196)(35 68)(37 87)(38 200)(39 72)(41 91)(42 204)(43 52)(45 95)(46 208)(47 56)(49 108)(50 62)(51 92)(53 112)(54 66)(55 96)(57 116)(58 70)(59 76)(61 120)(63 80)(65 100)(67 84)(69 104)(71 88)(73 103)(74 198)(77 107)(78 202)(81 111)(82 206)(85 115)(86 210)(89 119)(90 214)(93 99)(94 194)(97 215)(101 195)(105 199)(109 203)(113 207)(117 211)(122 180)(123 229)(124 136)(126 184)(127 233)(128 140)(130 188)(131 237)(132 144)(134 192)(135 217)(138 172)(139 221)(142 176)(143 225)(145 157)(147 238)(148 191)(149 161)(151 218)(152 171)(153 165)(155 222)(156 175)(159 226)(160 179)(163 230)(164 183)(167 234)(168 187)(193 205)(197 209)(201 213)
(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 87 7 93 13 75 19 81)(2 86 20 80 14 74 8 92)(3 73 9 79 15 85 21 91)(4 96 22 90 16 84 10 78)(5 83 11 89 17 95 23 77)(6 82 24 76 18 94 12 88)(25 228 111 174 37 240 99 186)(26 191 100 221 38 179 112 233)(27 238 113 184 39 226 101 172)(28 177 102 231 40 189 114 219)(29 224 115 170 41 236 103 182)(30 187 104 217 42 175 116 229)(31 234 117 180 43 222 105 192)(32 173 106 227 44 185 118 239)(33 220 119 190 45 232 107 178)(34 183 108 237 46 171 120 225)(35 230 97 176 47 218 109 188)(36 169 110 223 48 181 98 235)(49 131 208 152 61 143 196 164)(50 145 197 124 62 157 209 136)(51 141 210 162 63 129 198 150)(52 155 199 134 64 167 211 122)(53 127 212 148 65 139 200 160)(54 165 201 144 66 153 213 132)(55 137 214 158 67 125 202 146)(56 151 203 130 68 163 215 142)(57 123 216 168 69 135 204 156)(58 161 205 140 70 149 193 128)(59 133 194 154 71 121 206 166)(60 147 207 126 72 159 195 138)

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

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

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

36 conjugacy classes

class 1 2A2B 3 4A4B4C 5 6A6B8A8B8C8D8E8F8G8H 10 12A12B12C 15 20A20B24A24B24C24D 30 40A40B40C40D60A60B
order122344456688888888101212121520202424242430404040406060
size111022554220662020303060604410108442020202081212121288

36 irreducible representations

dim1111112222222244444888
type++++++-+-+++-
imageC1C2C2C2C4C4S3D4Q8D6C3⋊D4C4×S3Dic6C8.C4F5C2×F5C4⋊F5C12.53D4C40.C4S3×F5Dic3⋊F5D10.Dic6
kernelD10.Dic6D5×C3⋊C8C3×C4.F5C12.F5C5×C3⋊C8C153C8C4.F5C3×Dic5C6×D5C4×D5Dic5C20D10C15C3⋊C8C12C6C5C3C4C2C1
# reps1111221111222411224112

Matrix representation of D10.Dic6 in GL6(𝔽241)

100000
010000
00000240
001111
00240000
00024000
,
24000000
02400000
00000240
00002400
00024000
00240000
,
177640000
17700000
00211181214208
0033276030
003332146
00211323830
,
137340000
1711040000
0017807777
001641011640
000164101164
0077770178

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,240,0,0,0,0,1,0,240,0,0,0,1,0,0,0,0,240,1,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240,0,0,0],[177,177,0,0,0,0,64,0,0,0,0,0,0,0,211,33,33,211,0,0,181,27,3,3,0,0,214,60,214,238,0,0,208,30,6,30],[137,171,0,0,0,0,34,104,0,0,0,0,0,0,178,164,0,77,0,0,0,101,164,77,0,0,77,164,101,0,0,0,77,0,164,178] >;

D10.Dic6 in GAP, Magma, Sage, TeX

D_{10}.{\rm Dic}_6
% in TeX

G:=Group("D10.Dic6");
// GroupNames label

G:=SmallGroup(480,237);
// by ID

G=gap.SmallGroup(480,237);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,64,100,675,80,1356,9414,4724]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=1,c^12=a^5,d^2=a^-1*b*c^6,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d^-1=a^-1*b*c^11>;
// generators/relations

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