metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10.Dic3, D4.(C3⋊F5), (C3×D4).F5, C3⋊5(D4.F5), (C5×D4).Dic3, C5⋊D4.Dic3, C15⋊17(C8○D4), C60.37(C2×C4), (D4×C15).1C4, (C4×D5).39D6, C12.19(C2×F5), C60.C4⋊7C2, C12.F5⋊9C2, D4⋊2D5.3S3, C5⋊1(D4.Dic3), C20.5(C2×Dic3), C6.38(C22×F5), C15⋊8M4(2)⋊6C2, C30.76(C22×C4), C15⋊C8.7C22, D10.1(C2×Dic3), (C2×Dic5).77D6, (C3×Dic10).1C4, Dic5.1(C2×Dic3), (D5×C12).72C22, C10.7(C22×Dic3), Dic5.51(C22×S3), (C3×Dic5).65C23, (C6×Dic5).170C22, C4.5(C2×C3⋊F5), (C2×C6).9(C2×F5), C2.8(C22×C3⋊F5), C22.1(C2×C3⋊F5), (C2×C15⋊C8)⋊11C2, (C3×C5⋊D4).1C4, (C2×C10).(C2×Dic3), (C2×C30).24(C2×C4), (C6×D5).26(C2×C4), (C3×D4⋊2D5).3C2, (C3×Dic5).31(C2×C4), SmallGroup(480,1066)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10.Dic3
G = < a,b,c,d | a20=1, b2=c6=a10, d2=a10c3, bab-1=a-1, cac-1=a9, dad-1=a17, bc=cb, bd=db, dcd-1=c5 >
Subgroups: 460 in 124 conjugacy classes, 57 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, D4, D4, Q8, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, M4(2), C4○D4, Dic5, Dic5, C20, D10, C2×C10, C3⋊C8, C2×C12, C3×D4, C3×D4, C3×Q8, C3×D5, C30, C30, C8○D4, C5⋊C8, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C2×C3⋊C8, C4.Dic3, C3×C4○D4, C3×Dic5, C3×Dic5, C60, C6×D5, C2×C30, D5⋊C8, C4.F5, C2×C5⋊C8, C22.F5, D4⋊2D5, D4.Dic3, C15⋊C8, C15⋊C8, C3×Dic10, D5×C12, C6×Dic5, C3×C5⋊D4, D4×C15, D4.F5, C60.C4, C12.F5, C2×C15⋊C8, C15⋊8M4(2), C3×D4⋊2D5, Dic10.Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, F5, C2×Dic3, C22×S3, C8○D4, C2×F5, C22×Dic3, C3⋊F5, C22×F5, D4.Dic3, C2×C3⋊F5, D4.F5, C22×C3⋊F5, Dic10.Dic3
►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)
(1 124 11 134)(2 123 12 133)(3 122 13 132)(4 121 14 131)(5 140 15 130)(6 139 16 129)(7 138 17 128)(8 137 18 127)(9 136 19 126)(10 135 20 125)(21 106 31 116)(22 105 32 115)(23 104 33 114)(24 103 34 113)(25 102 35 112)(26 101 36 111)(27 120 37 110)(28 119 38 109)(29 118 39 108)(30 117 40 107)(41 177 51 167)(42 176 52 166)(43 175 53 165)(44 174 54 164)(45 173 55 163)(46 172 56 162)(47 171 57 161)(48 170 58 180)(49 169 59 179)(50 168 60 178)(61 238 71 228)(62 237 72 227)(63 236 73 226)(64 235 74 225)(65 234 75 224)(66 233 76 223)(67 232 77 222)(68 231 78 221)(69 230 79 240)(70 229 80 239)(81 150 91 160)(82 149 92 159)(83 148 93 158)(84 147 94 157)(85 146 95 156)(86 145 96 155)(87 144 97 154)(88 143 98 153)(89 142 99 152)(90 141 100 151)(181 215 191 205)(182 214 192 204)(183 213 193 203)(184 212 194 202)(185 211 195 201)(186 210 196 220)(187 209 197 219)(188 208 198 218)(189 207 199 217)(190 206 200 216)
(1 153 175 129 93 58 11 143 165 139 83 48)(2 142 176 138 94 47 12 152 166 128 84 57)(3 151 177 127 95 56 13 141 167 137 85 46)(4 160 178 136 96 45 14 150 168 126 86 55)(5 149 179 125 97 54 15 159 169 135 87 44)(6 158 180 134 98 43 16 148 170 124 88 53)(7 147 161 123 99 52 17 157 171 133 89 42)(8 156 162 132 100 41 18 146 172 122 90 51)(9 145 163 121 81 50 19 155 173 131 91 60)(10 154 164 130 82 59 20 144 174 140 92 49)(21 228 215 111 66 196 31 238 205 101 76 186)(22 237 216 120 67 185 32 227 206 110 77 195)(23 226 217 109 68 194 33 236 207 119 78 184)(24 235 218 118 69 183 34 225 208 108 79 193)(25 224 219 107 70 192 35 234 209 117 80 182)(26 233 220 116 71 181 36 223 210 106 61 191)(27 222 201 105 72 190 37 232 211 115 62 200)(28 231 202 114 73 199 38 221 212 104 63 189)(29 240 203 103 74 188 39 230 213 113 64 198)(30 229 204 112 75 197 40 239 214 102 65 187)
(1 228 139 76 11 238 129 66)(2 221 128 73 12 231 138 63)(3 234 137 70 13 224 127 80)(4 227 126 67 14 237 136 77)(5 240 135 64 15 230 125 74)(6 233 124 61 16 223 134 71)(7 226 133 78 17 236 123 68)(8 239 122 75 18 229 132 65)(9 232 131 72 19 222 121 62)(10 225 140 69 20 235 130 79)(21 93 101 153 31 83 111 143)(22 86 110 150 32 96 120 160)(23 99 119 147 33 89 109 157)(24 92 108 144 34 82 118 154)(25 85 117 141 35 95 107 151)(26 98 106 158 36 88 116 148)(27 91 115 155 37 81 105 145)(28 84 104 152 38 94 114 142)(29 97 113 149 39 87 103 159)(30 90 102 146 40 100 112 156)(41 214 162 197 51 204 172 187)(42 207 171 194 52 217 161 184)(43 220 180 191 53 210 170 181)(44 213 169 188 54 203 179 198)(45 206 178 185 55 216 168 195)(46 219 167 182 56 209 177 192)(47 212 176 199 57 202 166 189)(48 205 165 196 58 215 175 186)(49 218 174 193 59 208 164 183)(50 211 163 190 60 201 173 200)
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), (1,124,11,134)(2,123,12,133)(3,122,13,132)(4,121,14,131)(5,140,15,130)(6,139,16,129)(7,138,17,128)(8,137,18,127)(9,136,19,126)(10,135,20,125)(21,106,31,116)(22,105,32,115)(23,104,33,114)(24,103,34,113)(25,102,35,112)(26,101,36,111)(27,120,37,110)(28,119,38,109)(29,118,39,108)(30,117,40,107)(41,177,51,167)(42,176,52,166)(43,175,53,165)(44,174,54,164)(45,173,55,163)(46,172,56,162)(47,171,57,161)(48,170,58,180)(49,169,59,179)(50,168,60,178)(61,238,71,228)(62,237,72,227)(63,236,73,226)(64,235,74,225)(65,234,75,224)(66,233,76,223)(67,232,77,222)(68,231,78,221)(69,230,79,240)(70,229,80,239)(81,150,91,160)(82,149,92,159)(83,148,93,158)(84,147,94,157)(85,146,95,156)(86,145,96,155)(87,144,97,154)(88,143,98,153)(89,142,99,152)(90,141,100,151)(181,215,191,205)(182,214,192,204)(183,213,193,203)(184,212,194,202)(185,211,195,201)(186,210,196,220)(187,209,197,219)(188,208,198,218)(189,207,199,217)(190,206,200,216), (1,153,175,129,93,58,11,143,165,139,83,48)(2,142,176,138,94,47,12,152,166,128,84,57)(3,151,177,127,95,56,13,141,167,137,85,46)(4,160,178,136,96,45,14,150,168,126,86,55)(5,149,179,125,97,54,15,159,169,135,87,44)(6,158,180,134,98,43,16,148,170,124,88,53)(7,147,161,123,99,52,17,157,171,133,89,42)(8,156,162,132,100,41,18,146,172,122,90,51)(9,145,163,121,81,50,19,155,173,131,91,60)(10,154,164,130,82,59,20,144,174,140,92,49)(21,228,215,111,66,196,31,238,205,101,76,186)(22,237,216,120,67,185,32,227,206,110,77,195)(23,226,217,109,68,194,33,236,207,119,78,184)(24,235,218,118,69,183,34,225,208,108,79,193)(25,224,219,107,70,192,35,234,209,117,80,182)(26,233,220,116,71,181,36,223,210,106,61,191)(27,222,201,105,72,190,37,232,211,115,62,200)(28,231,202,114,73,199,38,221,212,104,63,189)(29,240,203,103,74,188,39,230,213,113,64,198)(30,229,204,112,75,197,40,239,214,102,65,187), (1,228,139,76,11,238,129,66)(2,221,128,73,12,231,138,63)(3,234,137,70,13,224,127,80)(4,227,126,67,14,237,136,77)(5,240,135,64,15,230,125,74)(6,233,124,61,16,223,134,71)(7,226,133,78,17,236,123,68)(8,239,122,75,18,229,132,65)(9,232,131,72,19,222,121,62)(10,225,140,69,20,235,130,79)(21,93,101,153,31,83,111,143)(22,86,110,150,32,96,120,160)(23,99,119,147,33,89,109,157)(24,92,108,144,34,82,118,154)(25,85,117,141,35,95,107,151)(26,98,106,158,36,88,116,148)(27,91,115,155,37,81,105,145)(28,84,104,152,38,94,114,142)(29,97,113,149,39,87,103,159)(30,90,102,146,40,100,112,156)(41,214,162,197,51,204,172,187)(42,207,171,194,52,217,161,184)(43,220,180,191,53,210,170,181)(44,213,169,188,54,203,179,198)(45,206,178,185,55,216,168,195)(46,219,167,182,56,209,177,192)(47,212,176,199,57,202,166,189)(48,205,165,196,58,215,175,186)(49,218,174,193,59,208,164,183)(50,211,163,190,60,201,173,200)>;
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), (1,124,11,134)(2,123,12,133)(3,122,13,132)(4,121,14,131)(5,140,15,130)(6,139,16,129)(7,138,17,128)(8,137,18,127)(9,136,19,126)(10,135,20,125)(21,106,31,116)(22,105,32,115)(23,104,33,114)(24,103,34,113)(25,102,35,112)(26,101,36,111)(27,120,37,110)(28,119,38,109)(29,118,39,108)(30,117,40,107)(41,177,51,167)(42,176,52,166)(43,175,53,165)(44,174,54,164)(45,173,55,163)(46,172,56,162)(47,171,57,161)(48,170,58,180)(49,169,59,179)(50,168,60,178)(61,238,71,228)(62,237,72,227)(63,236,73,226)(64,235,74,225)(65,234,75,224)(66,233,76,223)(67,232,77,222)(68,231,78,221)(69,230,79,240)(70,229,80,239)(81,150,91,160)(82,149,92,159)(83,148,93,158)(84,147,94,157)(85,146,95,156)(86,145,96,155)(87,144,97,154)(88,143,98,153)(89,142,99,152)(90,141,100,151)(181,215,191,205)(182,214,192,204)(183,213,193,203)(184,212,194,202)(185,211,195,201)(186,210,196,220)(187,209,197,219)(188,208,198,218)(189,207,199,217)(190,206,200,216), (1,153,175,129,93,58,11,143,165,139,83,48)(2,142,176,138,94,47,12,152,166,128,84,57)(3,151,177,127,95,56,13,141,167,137,85,46)(4,160,178,136,96,45,14,150,168,126,86,55)(5,149,179,125,97,54,15,159,169,135,87,44)(6,158,180,134,98,43,16,148,170,124,88,53)(7,147,161,123,99,52,17,157,171,133,89,42)(8,156,162,132,100,41,18,146,172,122,90,51)(9,145,163,121,81,50,19,155,173,131,91,60)(10,154,164,130,82,59,20,144,174,140,92,49)(21,228,215,111,66,196,31,238,205,101,76,186)(22,237,216,120,67,185,32,227,206,110,77,195)(23,226,217,109,68,194,33,236,207,119,78,184)(24,235,218,118,69,183,34,225,208,108,79,193)(25,224,219,107,70,192,35,234,209,117,80,182)(26,233,220,116,71,181,36,223,210,106,61,191)(27,222,201,105,72,190,37,232,211,115,62,200)(28,231,202,114,73,199,38,221,212,104,63,189)(29,240,203,103,74,188,39,230,213,113,64,198)(30,229,204,112,75,197,40,239,214,102,65,187), (1,228,139,76,11,238,129,66)(2,221,128,73,12,231,138,63)(3,234,137,70,13,224,127,80)(4,227,126,67,14,237,136,77)(5,240,135,64,15,230,125,74)(6,233,124,61,16,223,134,71)(7,226,133,78,17,236,123,68)(8,239,122,75,18,229,132,65)(9,232,131,72,19,222,121,62)(10,225,140,69,20,235,130,79)(21,93,101,153,31,83,111,143)(22,86,110,150,32,96,120,160)(23,99,119,147,33,89,109,157)(24,92,108,144,34,82,118,154)(25,85,117,141,35,95,107,151)(26,98,106,158,36,88,116,148)(27,91,115,155,37,81,105,145)(28,84,104,152,38,94,114,142)(29,97,113,149,39,87,103,159)(30,90,102,146,40,100,112,156)(41,214,162,197,51,204,172,187)(42,207,171,194,52,217,161,184)(43,220,180,191,53,210,170,181)(44,213,169,188,54,203,179,198)(45,206,178,185,55,216,168,195)(46,219,167,182,56,209,177,192)(47,212,176,199,57,202,166,189)(48,205,165,196,58,215,175,186)(49,218,174,193,59,208,164,183)(50,211,163,190,60,201,173,200) );
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)], [(1,124,11,134),(2,123,12,133),(3,122,13,132),(4,121,14,131),(5,140,15,130),(6,139,16,129),(7,138,17,128),(8,137,18,127),(9,136,19,126),(10,135,20,125),(21,106,31,116),(22,105,32,115),(23,104,33,114),(24,103,34,113),(25,102,35,112),(26,101,36,111),(27,120,37,110),(28,119,38,109),(29,118,39,108),(30,117,40,107),(41,177,51,167),(42,176,52,166),(43,175,53,165),(44,174,54,164),(45,173,55,163),(46,172,56,162),(47,171,57,161),(48,170,58,180),(49,169,59,179),(50,168,60,178),(61,238,71,228),(62,237,72,227),(63,236,73,226),(64,235,74,225),(65,234,75,224),(66,233,76,223),(67,232,77,222),(68,231,78,221),(69,230,79,240),(70,229,80,239),(81,150,91,160),(82,149,92,159),(83,148,93,158),(84,147,94,157),(85,146,95,156),(86,145,96,155),(87,144,97,154),(88,143,98,153),(89,142,99,152),(90,141,100,151),(181,215,191,205),(182,214,192,204),(183,213,193,203),(184,212,194,202),(185,211,195,201),(186,210,196,220),(187,209,197,219),(188,208,198,218),(189,207,199,217),(190,206,200,216)], [(1,153,175,129,93,58,11,143,165,139,83,48),(2,142,176,138,94,47,12,152,166,128,84,57),(3,151,177,127,95,56,13,141,167,137,85,46),(4,160,178,136,96,45,14,150,168,126,86,55),(5,149,179,125,97,54,15,159,169,135,87,44),(6,158,180,134,98,43,16,148,170,124,88,53),(7,147,161,123,99,52,17,157,171,133,89,42),(8,156,162,132,100,41,18,146,172,122,90,51),(9,145,163,121,81,50,19,155,173,131,91,60),(10,154,164,130,82,59,20,144,174,140,92,49),(21,228,215,111,66,196,31,238,205,101,76,186),(22,237,216,120,67,185,32,227,206,110,77,195),(23,226,217,109,68,194,33,236,207,119,78,184),(24,235,218,118,69,183,34,225,208,108,79,193),(25,224,219,107,70,192,35,234,209,117,80,182),(26,233,220,116,71,181,36,223,210,106,61,191),(27,222,201,105,72,190,37,232,211,115,62,200),(28,231,202,114,73,199,38,221,212,104,63,189),(29,240,203,103,74,188,39,230,213,113,64,198),(30,229,204,112,75,197,40,239,214,102,65,187)], [(1,228,139,76,11,238,129,66),(2,221,128,73,12,231,138,63),(3,234,137,70,13,224,127,80),(4,227,126,67,14,237,136,77),(5,240,135,64,15,230,125,74),(6,233,124,61,16,223,134,71),(7,226,133,78,17,236,123,68),(8,239,122,75,18,229,132,65),(9,232,131,72,19,222,121,62),(10,225,140,69,20,235,130,79),(21,93,101,153,31,83,111,143),(22,86,110,150,32,96,120,160),(23,99,119,147,33,89,109,157),(24,92,108,144,34,82,118,154),(25,85,117,141,35,95,107,151),(26,98,106,158,36,88,116,148),(27,91,115,155,37,81,105,145),(28,84,104,152,38,94,114,142),(29,97,113,149,39,87,103,159),(30,90,102,146,40,100,112,156),(41,214,162,197,51,204,172,187),(42,207,171,194,52,217,161,184),(43,220,180,191,53,210,170,181),(44,213,169,188,54,203,179,198),(45,206,178,185,55,216,168,195),(46,219,167,182,56,209,177,192),(47,212,176,199,57,202,166,189),(48,205,165,196,58,215,175,186),(49,218,174,193,59,208,164,183),(50,211,163,190,60,201,173,200)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5 | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20 | 30A | 30B | 30C | 30D | 30E | 30F | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 30 | 30 | 30 | 30 | 30 | 30 | 60 | 60 |
| size | 1 | 1 | 2 | 2 | 10 | 2 | 2 | 5 | 5 | 10 | 10 | 4 | 2 | 4 | 4 | 20 | 15 | 15 | 15 | 15 | 30 | ··· | 30 | 4 | 8 | 8 | 4 | 10 | 10 | 20 | 20 | 4 | 4 | 8 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | - | + | + | - | - | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | Dic3 | D6 | D6 | Dic3 | Dic3 | C8○D4 | F5 | C2×F5 | C2×F5 | C3⋊F5 | D4.Dic3 | C2×C3⋊F5 | C2×C3⋊F5 | D4.F5 | Dic10.Dic3 |
| kernel | Dic10.Dic3 | C60.C4 | C12.F5 | C2×C15⋊C8 | C15⋊8M4(2) | C3×D4⋊2D5 | C3×Dic10 | C3×C5⋊D4 | D4×C15 | D4⋊2D5 | Dic10 | C4×D5 | C2×Dic5 | C5⋊D4 | C5×D4 | C15 | C3×D4 | C12 | C2×C6 | D4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 2 |
Matrix representation of Dic10.Dic3 ►in GL8(𝔽241)
| 1 | 238 | 0 | 0 | 0 | 0 | 0 | 0 |
| 81 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 64 | 49 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 126 | 90 | 124 | 130 |
| 0 | 0 | 0 | 0 | 216 | 214 | 13 | 142 |
| 0 | 0 | 0 | 0 | 99 | 103 | 25 | 27 |
| 0 | 0 | 0 | 0 | 229 | 115 | 151 | 117 |
| 177 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 115 | 151 | 117 | 111 |
| 0 | 0 | 0 | 0 | 25 | 27 | 228 | 99 |
| 0 | 0 | 0 | 0 | 142 | 138 | 216 | 214 |
| 0 | 0 | 0 | 0 | 12 | 126 | 90 | 124 |
| 211 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 211 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 152 | 170 | 0 | 0 | 0 | 0 |
| 0 | 0 | 81 | 89 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 177 | 217 | 75 | 226 |
| 0 | 0 | 0 | 0 | 162 | 4 | 11 | 202 |
| 0 | 0 | 0 | 0 | 138 | 79 | 237 | 230 |
| 0 | 0 | 0 | 0 | 166 | 15 | 213 | 64 |
G:=sub<GL(8,GF(241))| [1,81,0,0,0,0,0,0,238,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,240,240,240,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[64,0,0,0,0,0,0,0,49,177,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,126,216,99,229,0,0,0,0,90,214,103,115,0,0,0,0,124,13,25,151,0,0,0,0,130,142,27,117],[177,0,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,115,25,142,12,0,0,0,0,151,27,138,126,0,0,0,0,117,228,216,90,0,0,0,0,111,99,214,124],[211,0,0,0,0,0,0,0,0,211,0,0,0,0,0,0,0,0,152,81,0,0,0,0,0,0,170,89,0,0,0,0,0,0,0,0,177,162,138,166,0,0,0,0,217,4,79,15,0,0,0,0,75,11,237,213,0,0,0,0,226,202,230,64] >;
Dic10.Dic3 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}.{\rm Dic}_3 % in TeX
G:=Group("Dic10.Dic3"); // GroupNames label
G:=SmallGroup(480,1066);
// by ID
G=gap.SmallGroup(480,1066);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,219,80,2693,14118,2379]);
// Polycyclic
G:=Group<a,b,c,d|a^20=1,b^2=c^6=a^10,d^2=a^10*c^3,b*a*b^-1=a^-1,c*a*c^-1=a^9,d*a*d^-1=a^17,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations