direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C15×C4.10D4, C60.244D4, M4(2).1C30, (C2×C4).C60, (C2×C60).26C4, (C2×C12).2C20, C4.10(D4×C15), C20.59(C3×D4), C12.59(C5×D4), (C6×Q8).6C10, (C2×Q8).3C30, (C2×C20).13C12, C22.4(C2×C60), (Q8×C10).10C6, (Q8×C30).16C2, (C5×M4(2)).3C6, (C2×C60).419C22, (C15×M4(2)).7C2, (C3×M4(2)).3C10, C30.127(C22⋊C4), (C2×C4).2(C2×C30), (C2×C20).60(C2×C6), (C2×C6).21(C2×C20), C2.5(C15×C22⋊C4), C6.23(C5×C22⋊C4), (C2×C10).41(C2×C12), (C2×C30).166(C2×C4), (C2×C12).60(C2×C10), C10.34(C3×C22⋊C4), SmallGroup(480,204)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C4.10D4
G = < a,b,c,d | a15=b4=1, c4=b2, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >
Subgroups: 104 in 76 conjugacy classes, 48 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, C10, C10, C12, C12, C2×C6, C15, M4(2), C2×Q8, C20, C20, C2×C10, C24, C2×C12, C2×C12, C3×Q8, C30, C30, C4.10D4, C40, C2×C20, C2×C20, C5×Q8, C3×M4(2), C6×Q8, C60, C60, C2×C30, C5×M4(2), Q8×C10, C3×C4.10D4, C120, C2×C60, C2×C60, Q8×C15, C5×C4.10D4, C15×M4(2), Q8×C30, C15×C4.10D4
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, D4, C10, C12, C2×C6, C15, C22⋊C4, C20, C2×C10, C2×C12, C3×D4, C30, C4.10D4, C2×C20, C5×D4, C3×C22⋊C4, C60, C2×C30, C5×C22⋊C4, C3×C4.10D4, C2×C60, D4×C15, C5×C4.10D4, C15×C22⋊C4, C15×C4.10D4
►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 63 103 36)(2 64 104 37)(3 65 105 38)(4 66 91 39)(5 67 92 40)(6 68 93 41)(7 69 94 42)(8 70 95 43)(9 71 96 44)(10 72 97 45)(11 73 98 31)(12 74 99 32)(13 75 100 33)(14 61 101 34)(15 62 102 35)(16 192 223 200)(17 193 224 201)(18 194 225 202)(19 195 211 203)(20 181 212 204)(21 182 213 205)(22 183 214 206)(23 184 215 207)(24 185 216 208)(25 186 217 209)(26 187 218 210)(27 188 219 196)(28 189 220 197)(29 190 221 198)(30 191 222 199)(46 153 108 130)(47 154 109 131)(48 155 110 132)(49 156 111 133)(50 157 112 134)(51 158 113 135)(52 159 114 121)(53 160 115 122)(54 161 116 123)(55 162 117 124)(56 163 118 125)(57 164 119 126)(58 165 120 127)(59 151 106 128)(60 152 107 129)(76 146 240 179)(77 147 226 180)(78 148 227 166)(79 149 228 167)(80 150 229 168)(81 136 230 169)(82 137 231 170)(83 138 232 171)(84 139 233 172)(85 140 234 173)(86 141 235 174)(87 142 236 175)(88 143 237 176)(89 144 238 177)(90 145 239 178)
(1 175 63 236 103 142 36 87)(2 176 64 237 104 143 37 88)(3 177 65 238 105 144 38 89)(4 178 66 239 91 145 39 90)(5 179 67 240 92 146 40 76)(6 180 68 226 93 147 41 77)(7 166 69 227 94 148 42 78)(8 167 70 228 95 149 43 79)(9 168 71 229 96 150 44 80)(10 169 72 230 97 136 45 81)(11 170 73 231 98 137 31 82)(12 171 74 232 99 138 32 83)(13 172 75 233 100 139 33 84)(14 173 61 234 101 140 34 85)(15 174 62 235 102 141 35 86)(16 60 192 129 223 107 200 152)(17 46 193 130 224 108 201 153)(18 47 194 131 225 109 202 154)(19 48 195 132 211 110 203 155)(20 49 181 133 212 111 204 156)(21 50 182 134 213 112 205 157)(22 51 183 135 214 113 206 158)(23 52 184 121 215 114 207 159)(24 53 185 122 216 115 208 160)(25 54 186 123 217 116 209 161)(26 55 187 124 218 117 210 162)(27 56 188 125 219 118 196 163)(28 57 189 126 220 119 197 164)(29 58 190 127 221 120 198 165)(30 59 191 128 222 106 199 151)
(1 54 36 123 103 116 63 161)(2 55 37 124 104 117 64 162)(3 56 38 125 105 118 65 163)(4 57 39 126 91 119 66 164)(5 58 40 127 92 120 67 165)(6 59 41 128 93 106 68 151)(7 60 42 129 94 107 69 152)(8 46 43 130 95 108 70 153)(9 47 44 131 96 109 71 154)(10 48 45 132 97 110 72 155)(11 49 31 133 98 111 73 156)(12 50 32 134 99 112 74 157)(13 51 33 135 100 113 75 158)(14 52 34 121 101 114 61 159)(15 53 35 122 102 115 62 160)(16 148 200 78 223 166 192 227)(17 149 201 79 224 167 193 228)(18 150 202 80 225 168 194 229)(19 136 203 81 211 169 195 230)(20 137 204 82 212 170 181 231)(21 138 205 83 213 171 182 232)(22 139 206 84 214 172 183 233)(23 140 207 85 215 173 184 234)(24 141 208 86 216 174 185 235)(25 142 209 87 217 175 186 236)(26 143 210 88 218 176 187 237)(27 144 196 89 219 177 188 238)(28 145 197 90 220 178 189 239)(29 146 198 76 221 179 190 240)(30 147 199 77 222 180 191 226)
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), (1,63,103,36)(2,64,104,37)(3,65,105,38)(4,66,91,39)(5,67,92,40)(6,68,93,41)(7,69,94,42)(8,70,95,43)(9,71,96,44)(10,72,97,45)(11,73,98,31)(12,74,99,32)(13,75,100,33)(14,61,101,34)(15,62,102,35)(16,192,223,200)(17,193,224,201)(18,194,225,202)(19,195,211,203)(20,181,212,204)(21,182,213,205)(22,183,214,206)(23,184,215,207)(24,185,216,208)(25,186,217,209)(26,187,218,210)(27,188,219,196)(28,189,220,197)(29,190,221,198)(30,191,222,199)(46,153,108,130)(47,154,109,131)(48,155,110,132)(49,156,111,133)(50,157,112,134)(51,158,113,135)(52,159,114,121)(53,160,115,122)(54,161,116,123)(55,162,117,124)(56,163,118,125)(57,164,119,126)(58,165,120,127)(59,151,106,128)(60,152,107,129)(76,146,240,179)(77,147,226,180)(78,148,227,166)(79,149,228,167)(80,150,229,168)(81,136,230,169)(82,137,231,170)(83,138,232,171)(84,139,233,172)(85,140,234,173)(86,141,235,174)(87,142,236,175)(88,143,237,176)(89,144,238,177)(90,145,239,178), (1,175,63,236,103,142,36,87)(2,176,64,237,104,143,37,88)(3,177,65,238,105,144,38,89)(4,178,66,239,91,145,39,90)(5,179,67,240,92,146,40,76)(6,180,68,226,93,147,41,77)(7,166,69,227,94,148,42,78)(8,167,70,228,95,149,43,79)(9,168,71,229,96,150,44,80)(10,169,72,230,97,136,45,81)(11,170,73,231,98,137,31,82)(12,171,74,232,99,138,32,83)(13,172,75,233,100,139,33,84)(14,173,61,234,101,140,34,85)(15,174,62,235,102,141,35,86)(16,60,192,129,223,107,200,152)(17,46,193,130,224,108,201,153)(18,47,194,131,225,109,202,154)(19,48,195,132,211,110,203,155)(20,49,181,133,212,111,204,156)(21,50,182,134,213,112,205,157)(22,51,183,135,214,113,206,158)(23,52,184,121,215,114,207,159)(24,53,185,122,216,115,208,160)(25,54,186,123,217,116,209,161)(26,55,187,124,218,117,210,162)(27,56,188,125,219,118,196,163)(28,57,189,126,220,119,197,164)(29,58,190,127,221,120,198,165)(30,59,191,128,222,106,199,151), (1,54,36,123,103,116,63,161)(2,55,37,124,104,117,64,162)(3,56,38,125,105,118,65,163)(4,57,39,126,91,119,66,164)(5,58,40,127,92,120,67,165)(6,59,41,128,93,106,68,151)(7,60,42,129,94,107,69,152)(8,46,43,130,95,108,70,153)(9,47,44,131,96,109,71,154)(10,48,45,132,97,110,72,155)(11,49,31,133,98,111,73,156)(12,50,32,134,99,112,74,157)(13,51,33,135,100,113,75,158)(14,52,34,121,101,114,61,159)(15,53,35,122,102,115,62,160)(16,148,200,78,223,166,192,227)(17,149,201,79,224,167,193,228)(18,150,202,80,225,168,194,229)(19,136,203,81,211,169,195,230)(20,137,204,82,212,170,181,231)(21,138,205,83,213,171,182,232)(22,139,206,84,214,172,183,233)(23,140,207,85,215,173,184,234)(24,141,208,86,216,174,185,235)(25,142,209,87,217,175,186,236)(26,143,210,88,218,176,187,237)(27,144,196,89,219,177,188,238)(28,145,197,90,220,178,189,239)(29,146,198,76,221,179,190,240)(30,147,199,77,222,180,191,226)>;
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), (1,63,103,36)(2,64,104,37)(3,65,105,38)(4,66,91,39)(5,67,92,40)(6,68,93,41)(7,69,94,42)(8,70,95,43)(9,71,96,44)(10,72,97,45)(11,73,98,31)(12,74,99,32)(13,75,100,33)(14,61,101,34)(15,62,102,35)(16,192,223,200)(17,193,224,201)(18,194,225,202)(19,195,211,203)(20,181,212,204)(21,182,213,205)(22,183,214,206)(23,184,215,207)(24,185,216,208)(25,186,217,209)(26,187,218,210)(27,188,219,196)(28,189,220,197)(29,190,221,198)(30,191,222,199)(46,153,108,130)(47,154,109,131)(48,155,110,132)(49,156,111,133)(50,157,112,134)(51,158,113,135)(52,159,114,121)(53,160,115,122)(54,161,116,123)(55,162,117,124)(56,163,118,125)(57,164,119,126)(58,165,120,127)(59,151,106,128)(60,152,107,129)(76,146,240,179)(77,147,226,180)(78,148,227,166)(79,149,228,167)(80,150,229,168)(81,136,230,169)(82,137,231,170)(83,138,232,171)(84,139,233,172)(85,140,234,173)(86,141,235,174)(87,142,236,175)(88,143,237,176)(89,144,238,177)(90,145,239,178), (1,175,63,236,103,142,36,87)(2,176,64,237,104,143,37,88)(3,177,65,238,105,144,38,89)(4,178,66,239,91,145,39,90)(5,179,67,240,92,146,40,76)(6,180,68,226,93,147,41,77)(7,166,69,227,94,148,42,78)(8,167,70,228,95,149,43,79)(9,168,71,229,96,150,44,80)(10,169,72,230,97,136,45,81)(11,170,73,231,98,137,31,82)(12,171,74,232,99,138,32,83)(13,172,75,233,100,139,33,84)(14,173,61,234,101,140,34,85)(15,174,62,235,102,141,35,86)(16,60,192,129,223,107,200,152)(17,46,193,130,224,108,201,153)(18,47,194,131,225,109,202,154)(19,48,195,132,211,110,203,155)(20,49,181,133,212,111,204,156)(21,50,182,134,213,112,205,157)(22,51,183,135,214,113,206,158)(23,52,184,121,215,114,207,159)(24,53,185,122,216,115,208,160)(25,54,186,123,217,116,209,161)(26,55,187,124,218,117,210,162)(27,56,188,125,219,118,196,163)(28,57,189,126,220,119,197,164)(29,58,190,127,221,120,198,165)(30,59,191,128,222,106,199,151), (1,54,36,123,103,116,63,161)(2,55,37,124,104,117,64,162)(3,56,38,125,105,118,65,163)(4,57,39,126,91,119,66,164)(5,58,40,127,92,120,67,165)(6,59,41,128,93,106,68,151)(7,60,42,129,94,107,69,152)(8,46,43,130,95,108,70,153)(9,47,44,131,96,109,71,154)(10,48,45,132,97,110,72,155)(11,49,31,133,98,111,73,156)(12,50,32,134,99,112,74,157)(13,51,33,135,100,113,75,158)(14,52,34,121,101,114,61,159)(15,53,35,122,102,115,62,160)(16,148,200,78,223,166,192,227)(17,149,201,79,224,167,193,228)(18,150,202,80,225,168,194,229)(19,136,203,81,211,169,195,230)(20,137,204,82,212,170,181,231)(21,138,205,83,213,171,182,232)(22,139,206,84,214,172,183,233)(23,140,207,85,215,173,184,234)(24,141,208,86,216,174,185,235)(25,142,209,87,217,175,186,236)(26,143,210,88,218,176,187,237)(27,144,196,89,219,177,188,238)(28,145,197,90,220,178,189,239)(29,146,198,76,221,179,190,240)(30,147,199,77,222,180,191,226) );
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)], [(1,63,103,36),(2,64,104,37),(3,65,105,38),(4,66,91,39),(5,67,92,40),(6,68,93,41),(7,69,94,42),(8,70,95,43),(9,71,96,44),(10,72,97,45),(11,73,98,31),(12,74,99,32),(13,75,100,33),(14,61,101,34),(15,62,102,35),(16,192,223,200),(17,193,224,201),(18,194,225,202),(19,195,211,203),(20,181,212,204),(21,182,213,205),(22,183,214,206),(23,184,215,207),(24,185,216,208),(25,186,217,209),(26,187,218,210),(27,188,219,196),(28,189,220,197),(29,190,221,198),(30,191,222,199),(46,153,108,130),(47,154,109,131),(48,155,110,132),(49,156,111,133),(50,157,112,134),(51,158,113,135),(52,159,114,121),(53,160,115,122),(54,161,116,123),(55,162,117,124),(56,163,118,125),(57,164,119,126),(58,165,120,127),(59,151,106,128),(60,152,107,129),(76,146,240,179),(77,147,226,180),(78,148,227,166),(79,149,228,167),(80,150,229,168),(81,136,230,169),(82,137,231,170),(83,138,232,171),(84,139,233,172),(85,140,234,173),(86,141,235,174),(87,142,236,175),(88,143,237,176),(89,144,238,177),(90,145,239,178)], [(1,175,63,236,103,142,36,87),(2,176,64,237,104,143,37,88),(3,177,65,238,105,144,38,89),(4,178,66,239,91,145,39,90),(5,179,67,240,92,146,40,76),(6,180,68,226,93,147,41,77),(7,166,69,227,94,148,42,78),(8,167,70,228,95,149,43,79),(9,168,71,229,96,150,44,80),(10,169,72,230,97,136,45,81),(11,170,73,231,98,137,31,82),(12,171,74,232,99,138,32,83),(13,172,75,233,100,139,33,84),(14,173,61,234,101,140,34,85),(15,174,62,235,102,141,35,86),(16,60,192,129,223,107,200,152),(17,46,193,130,224,108,201,153),(18,47,194,131,225,109,202,154),(19,48,195,132,211,110,203,155),(20,49,181,133,212,111,204,156),(21,50,182,134,213,112,205,157),(22,51,183,135,214,113,206,158),(23,52,184,121,215,114,207,159),(24,53,185,122,216,115,208,160),(25,54,186,123,217,116,209,161),(26,55,187,124,218,117,210,162),(27,56,188,125,219,118,196,163),(28,57,189,126,220,119,197,164),(29,58,190,127,221,120,198,165),(30,59,191,128,222,106,199,151)], [(1,54,36,123,103,116,63,161),(2,55,37,124,104,117,64,162),(3,56,38,125,105,118,65,163),(4,57,39,126,91,119,66,164),(5,58,40,127,92,120,67,165),(6,59,41,128,93,106,68,151),(7,60,42,129,94,107,69,152),(8,46,43,130,95,108,70,153),(9,47,44,131,96,109,71,154),(10,48,45,132,97,110,72,155),(11,49,31,133,98,111,73,156),(12,50,32,134,99,112,74,157),(13,51,33,135,100,113,75,158),(14,52,34,121,101,114,61,159),(15,53,35,122,102,115,62,160),(16,148,200,78,223,166,192,227),(17,149,201,79,224,167,193,228),(18,150,202,80,225,168,194,229),(19,136,203,81,211,169,195,230),(20,137,204,82,212,170,181,231),(21,138,205,83,213,171,182,232),(22,139,206,84,214,172,183,233),(23,140,207,85,215,173,184,234),(24,141,208,86,216,174,185,235),(25,142,209,87,217,175,186,236),(26,143,210,88,218,176,187,237),(27,144,196,89,219,177,188,238),(28,145,197,90,220,178,189,239),(29,146,198,76,221,179,190,240),(30,147,199,77,222,180,191,226)]])
165 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | ··· | 15H | 20A | ··· | 20H | 20I | ··· | 20P | 24A | ··· | 24H | 30A | ··· | 30H | 30I | ··· | 30P | 40A | ··· | 40P | 60A | ··· | 60P | 60Q | ··· | 60AF | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
165 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | |||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C5 | C6 | C6 | C10 | C10 | C12 | C15 | C20 | C30 | C30 | C60 | D4 | C3×D4 | C5×D4 | D4×C15 | C4.10D4 | C3×C4.10D4 | C5×C4.10D4 | C15×C4.10D4 |
| kernel | C15×C4.10D4 | C15×M4(2) | Q8×C30 | C5×C4.10D4 | C2×C60 | C3×C4.10D4 | C5×M4(2) | Q8×C10 | C3×M4(2) | C6×Q8 | C2×C20 | C4.10D4 | C2×C12 | M4(2) | C2×Q8 | C2×C4 | C60 | C20 | C12 | C4 | C15 | C5 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 8 | 4 | 8 | 8 | 16 | 16 | 8 | 32 | 2 | 4 | 8 | 16 | 1 | 2 | 4 | 8 |
Matrix representation of C15×C4.10D4 ►in GL6(𝔽241)
| 225 | 0 | 0 | 0 | 0 | 0 |
| 0 | 225 | 0 | 0 | 0 | 0 |
| 0 | 0 | 98 | 0 | 0 | 0 |
| 0 | 0 | 0 | 98 | 0 | 0 |
| 0 | 0 | 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 0 | 0 | 98 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 74 | 0 |
| 0 | 0 | 240 | 0 | 74 | 167 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 0 | 0 | 0 | 239 | 1 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 167 | 0 | 1 | 33 |
| 0 | 0 | 167 | 0 | 0 | 33 |
| 0 | 0 | 1 | 1 | 0 | 74 |
| 0 | 0 | 2 | 0 | 0 | 74 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 218 | 35 | 29 | 1 |
| 0 | 0 | 218 | 35 | 134 | 30 |
| 0 | 0 | 78 | 136 | 0 | 206 |
| 0 | 0 | 183 | 214 | 0 | 229 |
G:=sub<GL(6,GF(241))| [225,0,0,0,0,0,0,225,0,0,0,0,0,0,98,0,0,0,0,0,0,98,0,0,0,0,0,0,98,0,0,0,0,0,0,98],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,1,0,0,0,0,0,74,74,240,239,0,0,0,167,1,1],[0,1,0,0,0,0,240,0,0,0,0,0,0,0,167,167,1,2,0,0,0,0,1,0,0,0,1,0,0,0,0,0,33,33,74,74],[0,240,0,0,0,0,240,0,0,0,0,0,0,0,218,218,78,183,0,0,35,35,136,214,0,0,29,134,0,0,0,0,1,30,206,229] >;
C15×C4.10D4 in GAP, Magma, Sage, TeX
C_{15}\times C_4._{10}D_4 % in TeX
G:=Group("C15xC4.10D4"); // GroupNames label
G:=SmallGroup(480,204);
// by ID
G=gap.SmallGroup(480,204);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,840,869,1688,10504,7572,124]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^4=1,c^4=b^2,d^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations