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), C12.F5⋊9C2, C60.C4⋊7C2, 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, (C2×Dic5).77D6, D10.1(C2×Dic3), (C3×Dic10).1C4, Dic5.1(C2×Dic3), (D5×C12).72C22, C10.7(C22×Dic3), (C3×Dic5).65C23, Dic5.51(C22×S3), (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
Subgroups: 460 in 124 conjugacy classes, 57 normal (31 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×2], C22, C5, C6, C6 [×3], C8 [×4], C2×C4 [×3], D4, D4 [×2], Q8, D5, C10, C10 [×2], C12, C12 [×3], C2×C6 [×2], C2×C6, C15, C2×C8 [×3], M4(2) [×3], C4○D4, Dic5, Dic5 [×2], C20, D10, C2×C10 [×2], C3⋊C8 [×4], C2×C12 [×3], C3×D4, C3×D4 [×2], C3×Q8, C3×D5, C30, C30 [×2], C8○D4, C5⋊C8 [×4], Dic10, C4×D5, C2×Dic5 [×2], C5⋊D4 [×2], C5×D4, C2×C3⋊C8 [×3], C4.Dic3 [×3], C3×C4○D4, C3×Dic5, C3×Dic5 [×2], C60, C6×D5, C2×C30 [×2], D5⋊C8, C4.F5, C2×C5⋊C8 [×2], C22.F5 [×2], D4⋊2D5, D4.Dic3, C15⋊C8 [×2], C15⋊C8 [×2], C3×Dic10, D5×C12, C6×Dic5 [×2], C3×C5⋊D4 [×2], D4×C15, D4.F5, C60.C4, C12.F5, C2×C15⋊C8 [×2], C15⋊8M4(2) [×2], C3×D4⋊2D5, Dic10.Dic3
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, Dic3 [×4], D6 [×3], C22×C4, F5, C2×Dic3 [×6], C22×S3, C8○D4, C2×F5 [×3], C22×Dic3, C3⋊F5, C22×F5, D4.Dic3, C2×C3⋊F5 [×3], D4.F5, C22×C3⋊F5, Dic10.Dic3
Generators and relations
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 >
►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)
(1 238 11 228)(2 237 12 227)(3 236 13 226)(4 235 14 225)(5 234 15 224)(6 233 16 223)(7 232 17 222)(8 231 18 221)(9 230 19 240)(10 229 20 239)(21 172 31 162)(22 171 32 161)(23 170 33 180)(24 169 34 179)(25 168 35 178)(26 167 36 177)(27 166 37 176)(28 165 38 175)(29 164 39 174)(30 163 40 173)(41 81 51 91)(42 100 52 90)(43 99 53 89)(44 98 54 88)(45 97 55 87)(46 96 56 86)(47 95 57 85)(48 94 58 84)(49 93 59 83)(50 92 60 82)(61 209 71 219)(62 208 72 218)(63 207 73 217)(64 206 74 216)(65 205 75 215)(66 204 76 214)(67 203 77 213)(68 202 78 212)(69 201 79 211)(70 220 80 210)(101 140 111 130)(102 139 112 129)(103 138 113 128)(104 137 114 127)(105 136 115 126)(106 135 116 125)(107 134 117 124)(108 133 118 123)(109 132 119 122)(110 131 120 121)(141 191 151 181)(142 190 152 200)(143 189 153 199)(144 188 154 198)(145 187 155 197)(146 186 156 196)(147 185 157 195)(148 184 158 194)(149 183 159 193)(150 182 160 192)
(1 116 89 223 130 48 11 106 99 233 140 58)(2 105 90 232 131 57 12 115 100 222 121 47)(3 114 91 221 132 46 13 104 81 231 122 56)(4 103 92 230 133 55 14 113 82 240 123 45)(5 112 93 239 134 44 15 102 83 229 124 54)(6 101 94 228 135 53 16 111 84 238 125 43)(7 110 95 237 136 42 17 120 85 227 126 52)(8 119 96 226 137 51 18 109 86 236 127 41)(9 108 97 235 138 60 19 118 87 225 128 50)(10 117 98 224 139 49 20 107 88 234 129 59)(21 72 158 177 203 199 31 62 148 167 213 189)(22 61 159 166 204 188 32 71 149 176 214 198)(23 70 160 175 205 197 33 80 150 165 215 187)(24 79 141 164 206 186 34 69 151 174 216 196)(25 68 142 173 207 195 35 78 152 163 217 185)(26 77 143 162 208 184 36 67 153 172 218 194)(27 66 144 171 209 193 37 76 154 161 219 183)(28 75 145 180 210 182 38 65 155 170 220 192)(29 64 146 169 211 191 39 74 156 179 201 181)(30 73 147 178 212 200 40 63 157 168 202 190)
(1 217 233 78 11 207 223 68)(2 210 222 75 12 220 232 65)(3 203 231 72 13 213 221 62)(4 216 240 69 14 206 230 79)(5 209 229 66 15 219 239 76)(6 202 238 63 16 212 228 73)(7 215 227 80 17 205 237 70)(8 208 236 77 18 218 226 67)(9 201 225 74 19 211 235 64)(10 214 234 71 20 204 224 61)(21 114 167 122 31 104 177 132)(22 107 176 139 32 117 166 129)(23 120 165 136 33 110 175 126)(24 113 174 133 34 103 164 123)(25 106 163 130 35 116 173 140)(26 119 172 127 36 109 162 137)(27 112 161 124 37 102 171 134)(28 105 170 121 38 115 180 131)(29 118 179 138 39 108 169 128)(30 111 168 135 40 101 178 125)(41 194 86 153 51 184 96 143)(42 187 95 150 52 197 85 160)(43 200 84 147 53 190 94 157)(44 193 93 144 54 183 83 154)(45 186 82 141 55 196 92 151)(46 199 91 158 56 189 81 148)(47 192 100 155 57 182 90 145)(48 185 89 152 58 195 99 142)(49 198 98 149 59 188 88 159)(50 191 87 146 60 181 97 156)
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), (1,238,11,228)(2,237,12,227)(3,236,13,226)(4,235,14,225)(5,234,15,224)(6,233,16,223)(7,232,17,222)(8,231,18,221)(9,230,19,240)(10,229,20,239)(21,172,31,162)(22,171,32,161)(23,170,33,180)(24,169,34,179)(25,168,35,178)(26,167,36,177)(27,166,37,176)(28,165,38,175)(29,164,39,174)(30,163,40,173)(41,81,51,91)(42,100,52,90)(43,99,53,89)(44,98,54,88)(45,97,55,87)(46,96,56,86)(47,95,57,85)(48,94,58,84)(49,93,59,83)(50,92,60,82)(61,209,71,219)(62,208,72,218)(63,207,73,217)(64,206,74,216)(65,205,75,215)(66,204,76,214)(67,203,77,213)(68,202,78,212)(69,201,79,211)(70,220,80,210)(101,140,111,130)(102,139,112,129)(103,138,113,128)(104,137,114,127)(105,136,115,126)(106,135,116,125)(107,134,117,124)(108,133,118,123)(109,132,119,122)(110,131,120,121)(141,191,151,181)(142,190,152,200)(143,189,153,199)(144,188,154,198)(145,187,155,197)(146,186,156,196)(147,185,157,195)(148,184,158,194)(149,183,159,193)(150,182,160,192), (1,116,89,223,130,48,11,106,99,233,140,58)(2,105,90,232,131,57,12,115,100,222,121,47)(3,114,91,221,132,46,13,104,81,231,122,56)(4,103,92,230,133,55,14,113,82,240,123,45)(5,112,93,239,134,44,15,102,83,229,124,54)(6,101,94,228,135,53,16,111,84,238,125,43)(7,110,95,237,136,42,17,120,85,227,126,52)(8,119,96,226,137,51,18,109,86,236,127,41)(9,108,97,235,138,60,19,118,87,225,128,50)(10,117,98,224,139,49,20,107,88,234,129,59)(21,72,158,177,203,199,31,62,148,167,213,189)(22,61,159,166,204,188,32,71,149,176,214,198)(23,70,160,175,205,197,33,80,150,165,215,187)(24,79,141,164,206,186,34,69,151,174,216,196)(25,68,142,173,207,195,35,78,152,163,217,185)(26,77,143,162,208,184,36,67,153,172,218,194)(27,66,144,171,209,193,37,76,154,161,219,183)(28,75,145,180,210,182,38,65,155,170,220,192)(29,64,146,169,211,191,39,74,156,179,201,181)(30,73,147,178,212,200,40,63,157,168,202,190), (1,217,233,78,11,207,223,68)(2,210,222,75,12,220,232,65)(3,203,231,72,13,213,221,62)(4,216,240,69,14,206,230,79)(5,209,229,66,15,219,239,76)(6,202,238,63,16,212,228,73)(7,215,227,80,17,205,237,70)(8,208,236,77,18,218,226,67)(9,201,225,74,19,211,235,64)(10,214,234,71,20,204,224,61)(21,114,167,122,31,104,177,132)(22,107,176,139,32,117,166,129)(23,120,165,136,33,110,175,126)(24,113,174,133,34,103,164,123)(25,106,163,130,35,116,173,140)(26,119,172,127,36,109,162,137)(27,112,161,124,37,102,171,134)(28,105,170,121,38,115,180,131)(29,118,179,138,39,108,169,128)(30,111,168,135,40,101,178,125)(41,194,86,153,51,184,96,143)(42,187,95,150,52,197,85,160)(43,200,84,147,53,190,94,157)(44,193,93,144,54,183,83,154)(45,186,82,141,55,196,92,151)(46,199,91,158,56,189,81,148)(47,192,100,155,57,182,90,145)(48,185,89,152,58,195,99,142)(49,198,98,149,59,188,88,159)(50,191,87,146,60,181,97,156)>;
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), (1,238,11,228)(2,237,12,227)(3,236,13,226)(4,235,14,225)(5,234,15,224)(6,233,16,223)(7,232,17,222)(8,231,18,221)(9,230,19,240)(10,229,20,239)(21,172,31,162)(22,171,32,161)(23,170,33,180)(24,169,34,179)(25,168,35,178)(26,167,36,177)(27,166,37,176)(28,165,38,175)(29,164,39,174)(30,163,40,173)(41,81,51,91)(42,100,52,90)(43,99,53,89)(44,98,54,88)(45,97,55,87)(46,96,56,86)(47,95,57,85)(48,94,58,84)(49,93,59,83)(50,92,60,82)(61,209,71,219)(62,208,72,218)(63,207,73,217)(64,206,74,216)(65,205,75,215)(66,204,76,214)(67,203,77,213)(68,202,78,212)(69,201,79,211)(70,220,80,210)(101,140,111,130)(102,139,112,129)(103,138,113,128)(104,137,114,127)(105,136,115,126)(106,135,116,125)(107,134,117,124)(108,133,118,123)(109,132,119,122)(110,131,120,121)(141,191,151,181)(142,190,152,200)(143,189,153,199)(144,188,154,198)(145,187,155,197)(146,186,156,196)(147,185,157,195)(148,184,158,194)(149,183,159,193)(150,182,160,192), (1,116,89,223,130,48,11,106,99,233,140,58)(2,105,90,232,131,57,12,115,100,222,121,47)(3,114,91,221,132,46,13,104,81,231,122,56)(4,103,92,230,133,55,14,113,82,240,123,45)(5,112,93,239,134,44,15,102,83,229,124,54)(6,101,94,228,135,53,16,111,84,238,125,43)(7,110,95,237,136,42,17,120,85,227,126,52)(8,119,96,226,137,51,18,109,86,236,127,41)(9,108,97,235,138,60,19,118,87,225,128,50)(10,117,98,224,139,49,20,107,88,234,129,59)(21,72,158,177,203,199,31,62,148,167,213,189)(22,61,159,166,204,188,32,71,149,176,214,198)(23,70,160,175,205,197,33,80,150,165,215,187)(24,79,141,164,206,186,34,69,151,174,216,196)(25,68,142,173,207,195,35,78,152,163,217,185)(26,77,143,162,208,184,36,67,153,172,218,194)(27,66,144,171,209,193,37,76,154,161,219,183)(28,75,145,180,210,182,38,65,155,170,220,192)(29,64,146,169,211,191,39,74,156,179,201,181)(30,73,147,178,212,200,40,63,157,168,202,190), (1,217,233,78,11,207,223,68)(2,210,222,75,12,220,232,65)(3,203,231,72,13,213,221,62)(4,216,240,69,14,206,230,79)(5,209,229,66,15,219,239,76)(6,202,238,63,16,212,228,73)(7,215,227,80,17,205,237,70)(8,208,236,77,18,218,226,67)(9,201,225,74,19,211,235,64)(10,214,234,71,20,204,224,61)(21,114,167,122,31,104,177,132)(22,107,176,139,32,117,166,129)(23,120,165,136,33,110,175,126)(24,113,174,133,34,103,164,123)(25,106,163,130,35,116,173,140)(26,119,172,127,36,109,162,137)(27,112,161,124,37,102,171,134)(28,105,170,121,38,115,180,131)(29,118,179,138,39,108,169,128)(30,111,168,135,40,101,178,125)(41,194,86,153,51,184,96,143)(42,187,95,150,52,197,85,160)(43,200,84,147,53,190,94,157)(44,193,93,144,54,183,83,154)(45,186,82,141,55,196,92,151)(46,199,91,158,56,189,81,148)(47,192,100,155,57,182,90,145)(48,185,89,152,58,195,99,142)(49,198,98,149,59,188,88,159)(50,191,87,146,60,181,97,156) );
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)], [(1,238,11,228),(2,237,12,227),(3,236,13,226),(4,235,14,225),(5,234,15,224),(6,233,16,223),(7,232,17,222),(8,231,18,221),(9,230,19,240),(10,229,20,239),(21,172,31,162),(22,171,32,161),(23,170,33,180),(24,169,34,179),(25,168,35,178),(26,167,36,177),(27,166,37,176),(28,165,38,175),(29,164,39,174),(30,163,40,173),(41,81,51,91),(42,100,52,90),(43,99,53,89),(44,98,54,88),(45,97,55,87),(46,96,56,86),(47,95,57,85),(48,94,58,84),(49,93,59,83),(50,92,60,82),(61,209,71,219),(62,208,72,218),(63,207,73,217),(64,206,74,216),(65,205,75,215),(66,204,76,214),(67,203,77,213),(68,202,78,212),(69,201,79,211),(70,220,80,210),(101,140,111,130),(102,139,112,129),(103,138,113,128),(104,137,114,127),(105,136,115,126),(106,135,116,125),(107,134,117,124),(108,133,118,123),(109,132,119,122),(110,131,120,121),(141,191,151,181),(142,190,152,200),(143,189,153,199),(144,188,154,198),(145,187,155,197),(146,186,156,196),(147,185,157,195),(148,184,158,194),(149,183,159,193),(150,182,160,192)], [(1,116,89,223,130,48,11,106,99,233,140,58),(2,105,90,232,131,57,12,115,100,222,121,47),(3,114,91,221,132,46,13,104,81,231,122,56),(4,103,92,230,133,55,14,113,82,240,123,45),(5,112,93,239,134,44,15,102,83,229,124,54),(6,101,94,228,135,53,16,111,84,238,125,43),(7,110,95,237,136,42,17,120,85,227,126,52),(8,119,96,226,137,51,18,109,86,236,127,41),(9,108,97,235,138,60,19,118,87,225,128,50),(10,117,98,224,139,49,20,107,88,234,129,59),(21,72,158,177,203,199,31,62,148,167,213,189),(22,61,159,166,204,188,32,71,149,176,214,198),(23,70,160,175,205,197,33,80,150,165,215,187),(24,79,141,164,206,186,34,69,151,174,216,196),(25,68,142,173,207,195,35,78,152,163,217,185),(26,77,143,162,208,184,36,67,153,172,218,194),(27,66,144,171,209,193,37,76,154,161,219,183),(28,75,145,180,210,182,38,65,155,170,220,192),(29,64,146,169,211,191,39,74,156,179,201,181),(30,73,147,178,212,200,40,63,157,168,202,190)], [(1,217,233,78,11,207,223,68),(2,210,222,75,12,220,232,65),(3,203,231,72,13,213,221,62),(4,216,240,69,14,206,230,79),(5,209,229,66,15,219,239,76),(6,202,238,63,16,212,228,73),(7,215,227,80,17,205,237,70),(8,208,236,77,18,218,226,67),(9,201,225,74,19,211,235,64),(10,214,234,71,20,204,224,61),(21,114,167,122,31,104,177,132),(22,107,176,139,32,117,166,129),(23,120,165,136,33,110,175,126),(24,113,174,133,34,103,164,123),(25,106,163,130,35,116,173,140),(26,119,172,127,36,109,162,137),(27,112,161,124,37,102,171,134),(28,105,170,121,38,115,180,131),(29,118,179,138,39,108,169,128),(30,111,168,135,40,101,178,125),(41,194,86,153,51,184,96,143),(42,187,95,150,52,197,85,160),(43,200,84,147,53,190,94,157),(44,193,93,144,54,183,83,154),(45,186,82,141,55,196,92,151),(46,199,91,158,56,189,81,148),(47,192,100,155,57,182,90,145),(48,185,89,152,58,195,99,142),(49,198,98,149,59,188,88,159),(50,191,87,146,60,181,97,156)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
Dic_{10}.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
×
⋊
ℤ
𝔽
○
≀