direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C15×C4○D8, D8⋊3C30, Q16⋊3C30, SD16⋊3C30, C60.254D4, C60.295C23, C120.112C22, (C2×C8)⋊4C30, (C5×D8)⋊7C6, (C2×C40)⋊12C6, (C2×C24)⋊9C10, C4○D4⋊3C30, (C3×D8)⋊7C10, C8.6(C2×C30), (C5×Q16)⋊7C6, (C2×C120)⋊25C2, (C15×D8)⋊15C2, C40.28(C2×C6), (C3×Q16)⋊7C10, (C5×SD16)⋊7C6, D4.2(C2×C30), C4.20(D4×C15), C6.77(D4×C10), C12.69(C5×D4), C10.77(C6×D4), C20.69(C3×D4), C2.14(D4×C30), (C2×C30).98D4, Q8.5(C2×C30), C24.28(C2×C10), (C15×Q16)⋊15C2, (C3×SD16)⋊7C10, C30.460(C2×D4), C4.4(C22×C30), C22.1(D4×C15), (C15×SD16)⋊15C2, C20.47(C22×C6), (C2×C60).585C22, C12.47(C22×C10), (D4×C15).51C22, (Q8×C15).56C22, (C3×C4○D4)⋊6C10, (C5×C4○D4)⋊10C6, (C2×C6).11(C5×D4), (C15×C4○D4)⋊16C2, (C2×C4).29(C2×C30), (C5×D4).12(C2×C6), (C2×C10).11(C3×D4), (C5×Q8).21(C2×C6), (C2×C20).131(C2×C6), (C3×D4).12(C2×C10), (C3×Q8).13(C2×C10), (C2×C12).132(C2×C10), SmallGroup(480,940)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C4○D8
G = < a,b,c,d | a15=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >
Subgroups: 184 in 124 conjugacy classes, 80 normal (48 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, C20, C20, C2×C10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C30, C30, C4○D8, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C2×C24, C3×D8, C3×SD16, C3×Q16, C3×C4○D4, C60, C60, C2×C30, C2×C30, C2×C40, C5×D8, C5×SD16, C5×Q16, C5×C4○D4, C3×C4○D8, C120, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, C5×C4○D8, C2×C120, C15×D8, C15×SD16, C15×Q16, C15×C4○D4, C15×C4○D8
Quotients: C1, C2, C3, C22, C5, C6, D4, C23, C10, C2×C6, C15, C2×D4, C2×C10, C3×D4, C22×C6, C30, C4○D8, C5×D4, C22×C10, C6×D4, C2×C30, D4×C10, C3×C4○D8, D4×C15, C22×C30, C5×C4○D8, D4×C30, C15×C4○D8
►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 73 145 99)(2 74 146 100)(3 75 147 101)(4 61 148 102)(5 62 149 103)(6 63 150 104)(7 64 136 105)(8 65 137 91)(9 66 138 92)(10 67 139 93)(11 68 140 94)(12 69 141 95)(13 70 142 96)(14 71 143 97)(15 72 144 98)(16 169 127 199)(17 170 128 200)(18 171 129 201)(19 172 130 202)(20 173 131 203)(21 174 132 204)(22 175 133 205)(23 176 134 206)(24 177 135 207)(25 178 121 208)(26 179 122 209)(27 180 123 210)(28 166 124 196)(29 167 125 197)(30 168 126 198)(31 118 219 238)(32 119 220 239)(33 120 221 240)(34 106 222 226)(35 107 223 227)(36 108 224 228)(37 109 225 229)(38 110 211 230)(39 111 212 231)(40 112 213 232)(41 113 214 233)(42 114 215 234)(43 115 216 235)(44 116 217 236)(45 117 218 237)(46 192 84 165)(47 193 85 151)(48 194 86 152)(49 195 87 153)(50 181 88 154)(51 182 89 155)(52 183 90 156)(53 184 76 157)(54 185 77 158)(55 186 78 159)(56 187 79 160)(57 188 80 161)(58 189 81 162)(59 190 82 163)(60 191 83 164)
(1 45 73 117 145 218 99 237)(2 31 74 118 146 219 100 238)(3 32 75 119 147 220 101 239)(4 33 61 120 148 221 102 240)(5 34 62 106 149 222 103 226)(6 35 63 107 150 223 104 227)(7 36 64 108 136 224 105 228)(8 37 65 109 137 225 91 229)(9 38 66 110 138 211 92 230)(10 39 67 111 139 212 93 231)(11 40 68 112 140 213 94 232)(12 41 69 113 141 214 95 233)(13 42 70 114 142 215 96 234)(14 43 71 115 143 216 97 235)(15 44 72 116 144 217 98 236)(16 152 199 86 127 194 169 48)(17 153 200 87 128 195 170 49)(18 154 201 88 129 181 171 50)(19 155 202 89 130 182 172 51)(20 156 203 90 131 183 173 52)(21 157 204 76 132 184 174 53)(22 158 205 77 133 185 175 54)(23 159 206 78 134 186 176 55)(24 160 207 79 135 187 177 56)(25 161 208 80 121 188 178 57)(26 162 209 81 122 189 179 58)(27 163 210 82 123 190 180 59)(28 164 196 83 124 191 166 60)(29 165 197 84 125 192 167 46)(30 151 198 85 126 193 168 47)
(1 203)(2 204)(3 205)(4 206)(5 207)(6 208)(7 209)(8 210)(9 196)(10 197)(11 198)(12 199)(13 200)(14 201)(15 202)(16 69)(17 70)(18 71)(19 72)(20 73)(21 74)(22 75)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 157)(32 158)(33 159)(34 160)(35 161)(36 162)(37 163)(38 164)(39 165)(40 151)(41 152)(42 153)(43 154)(44 155)(45 156)(46 111)(47 112)(48 113)(49 114)(50 115)(51 116)(52 117)(53 118)(54 119)(55 120)(56 106)(57 107)(58 108)(59 109)(60 110)(76 238)(77 239)(78 240)(79 226)(80 227)(81 228)(82 229)(83 230)(84 231)(85 232)(86 233)(87 234)(88 235)(89 236)(90 237)(91 123)(92 124)(93 125)(94 126)(95 127)(96 128)(97 129)(98 130)(99 131)(100 132)(101 133)(102 134)(103 135)(104 121)(105 122)(136 179)(137 180)(138 166)(139 167)(140 168)(141 169)(142 170)(143 171)(144 172)(145 173)(146 174)(147 175)(148 176)(149 177)(150 178)(181 216)(182 217)(183 218)(184 219)(185 220)(186 221)(187 222)(188 223)(189 224)(190 225)(191 211)(192 212)(193 213)(194 214)(195 215)
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), (1,73,145,99)(2,74,146,100)(3,75,147,101)(4,61,148,102)(5,62,149,103)(6,63,150,104)(7,64,136,105)(8,65,137,91)(9,66,138,92)(10,67,139,93)(11,68,140,94)(12,69,141,95)(13,70,142,96)(14,71,143,97)(15,72,144,98)(16,169,127,199)(17,170,128,200)(18,171,129,201)(19,172,130,202)(20,173,131,203)(21,174,132,204)(22,175,133,205)(23,176,134,206)(24,177,135,207)(25,178,121,208)(26,179,122,209)(27,180,123,210)(28,166,124,196)(29,167,125,197)(30,168,126,198)(31,118,219,238)(32,119,220,239)(33,120,221,240)(34,106,222,226)(35,107,223,227)(36,108,224,228)(37,109,225,229)(38,110,211,230)(39,111,212,231)(40,112,213,232)(41,113,214,233)(42,114,215,234)(43,115,216,235)(44,116,217,236)(45,117,218,237)(46,192,84,165)(47,193,85,151)(48,194,86,152)(49,195,87,153)(50,181,88,154)(51,182,89,155)(52,183,90,156)(53,184,76,157)(54,185,77,158)(55,186,78,159)(56,187,79,160)(57,188,80,161)(58,189,81,162)(59,190,82,163)(60,191,83,164), (1,45,73,117,145,218,99,237)(2,31,74,118,146,219,100,238)(3,32,75,119,147,220,101,239)(4,33,61,120,148,221,102,240)(5,34,62,106,149,222,103,226)(6,35,63,107,150,223,104,227)(7,36,64,108,136,224,105,228)(8,37,65,109,137,225,91,229)(9,38,66,110,138,211,92,230)(10,39,67,111,139,212,93,231)(11,40,68,112,140,213,94,232)(12,41,69,113,141,214,95,233)(13,42,70,114,142,215,96,234)(14,43,71,115,143,216,97,235)(15,44,72,116,144,217,98,236)(16,152,199,86,127,194,169,48)(17,153,200,87,128,195,170,49)(18,154,201,88,129,181,171,50)(19,155,202,89,130,182,172,51)(20,156,203,90,131,183,173,52)(21,157,204,76,132,184,174,53)(22,158,205,77,133,185,175,54)(23,159,206,78,134,186,176,55)(24,160,207,79,135,187,177,56)(25,161,208,80,121,188,178,57)(26,162,209,81,122,189,179,58)(27,163,210,82,123,190,180,59)(28,164,196,83,124,191,166,60)(29,165,197,84,125,192,167,46)(30,151,198,85,126,193,168,47), (1,203)(2,204)(3,205)(4,206)(5,207)(6,208)(7,209)(8,210)(9,196)(10,197)(11,198)(12,199)(13,200)(14,201)(15,202)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,157)(32,158)(33,159)(34,160)(35,161)(36,162)(37,163)(38,164)(39,165)(40,151)(41,152)(42,153)(43,154)(44,155)(45,156)(46,111)(47,112)(48,113)(49,114)(50,115)(51,116)(52,117)(53,118)(54,119)(55,120)(56,106)(57,107)(58,108)(59,109)(60,110)(76,238)(77,239)(78,240)(79,226)(80,227)(81,228)(82,229)(83,230)(84,231)(85,232)(86,233)(87,234)(88,235)(89,236)(90,237)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128)(97,129)(98,130)(99,131)(100,132)(101,133)(102,134)(103,135)(104,121)(105,122)(136,179)(137,180)(138,166)(139,167)(140,168)(141,169)(142,170)(143,171)(144,172)(145,173)(146,174)(147,175)(148,176)(149,177)(150,178)(181,216)(182,217)(183,218)(184,219)(185,220)(186,221)(187,222)(188,223)(189,224)(190,225)(191,211)(192,212)(193,213)(194,214)(195,215)>;
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), (1,73,145,99)(2,74,146,100)(3,75,147,101)(4,61,148,102)(5,62,149,103)(6,63,150,104)(7,64,136,105)(8,65,137,91)(9,66,138,92)(10,67,139,93)(11,68,140,94)(12,69,141,95)(13,70,142,96)(14,71,143,97)(15,72,144,98)(16,169,127,199)(17,170,128,200)(18,171,129,201)(19,172,130,202)(20,173,131,203)(21,174,132,204)(22,175,133,205)(23,176,134,206)(24,177,135,207)(25,178,121,208)(26,179,122,209)(27,180,123,210)(28,166,124,196)(29,167,125,197)(30,168,126,198)(31,118,219,238)(32,119,220,239)(33,120,221,240)(34,106,222,226)(35,107,223,227)(36,108,224,228)(37,109,225,229)(38,110,211,230)(39,111,212,231)(40,112,213,232)(41,113,214,233)(42,114,215,234)(43,115,216,235)(44,116,217,236)(45,117,218,237)(46,192,84,165)(47,193,85,151)(48,194,86,152)(49,195,87,153)(50,181,88,154)(51,182,89,155)(52,183,90,156)(53,184,76,157)(54,185,77,158)(55,186,78,159)(56,187,79,160)(57,188,80,161)(58,189,81,162)(59,190,82,163)(60,191,83,164), (1,45,73,117,145,218,99,237)(2,31,74,118,146,219,100,238)(3,32,75,119,147,220,101,239)(4,33,61,120,148,221,102,240)(5,34,62,106,149,222,103,226)(6,35,63,107,150,223,104,227)(7,36,64,108,136,224,105,228)(8,37,65,109,137,225,91,229)(9,38,66,110,138,211,92,230)(10,39,67,111,139,212,93,231)(11,40,68,112,140,213,94,232)(12,41,69,113,141,214,95,233)(13,42,70,114,142,215,96,234)(14,43,71,115,143,216,97,235)(15,44,72,116,144,217,98,236)(16,152,199,86,127,194,169,48)(17,153,200,87,128,195,170,49)(18,154,201,88,129,181,171,50)(19,155,202,89,130,182,172,51)(20,156,203,90,131,183,173,52)(21,157,204,76,132,184,174,53)(22,158,205,77,133,185,175,54)(23,159,206,78,134,186,176,55)(24,160,207,79,135,187,177,56)(25,161,208,80,121,188,178,57)(26,162,209,81,122,189,179,58)(27,163,210,82,123,190,180,59)(28,164,196,83,124,191,166,60)(29,165,197,84,125,192,167,46)(30,151,198,85,126,193,168,47), (1,203)(2,204)(3,205)(4,206)(5,207)(6,208)(7,209)(8,210)(9,196)(10,197)(11,198)(12,199)(13,200)(14,201)(15,202)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,157)(32,158)(33,159)(34,160)(35,161)(36,162)(37,163)(38,164)(39,165)(40,151)(41,152)(42,153)(43,154)(44,155)(45,156)(46,111)(47,112)(48,113)(49,114)(50,115)(51,116)(52,117)(53,118)(54,119)(55,120)(56,106)(57,107)(58,108)(59,109)(60,110)(76,238)(77,239)(78,240)(79,226)(80,227)(81,228)(82,229)(83,230)(84,231)(85,232)(86,233)(87,234)(88,235)(89,236)(90,237)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128)(97,129)(98,130)(99,131)(100,132)(101,133)(102,134)(103,135)(104,121)(105,122)(136,179)(137,180)(138,166)(139,167)(140,168)(141,169)(142,170)(143,171)(144,172)(145,173)(146,174)(147,175)(148,176)(149,177)(150,178)(181,216)(182,217)(183,218)(184,219)(185,220)(186,221)(187,222)(188,223)(189,224)(190,225)(191,211)(192,212)(193,213)(194,214)(195,215) );
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)], [(1,73,145,99),(2,74,146,100),(3,75,147,101),(4,61,148,102),(5,62,149,103),(6,63,150,104),(7,64,136,105),(8,65,137,91),(9,66,138,92),(10,67,139,93),(11,68,140,94),(12,69,141,95),(13,70,142,96),(14,71,143,97),(15,72,144,98),(16,169,127,199),(17,170,128,200),(18,171,129,201),(19,172,130,202),(20,173,131,203),(21,174,132,204),(22,175,133,205),(23,176,134,206),(24,177,135,207),(25,178,121,208),(26,179,122,209),(27,180,123,210),(28,166,124,196),(29,167,125,197),(30,168,126,198),(31,118,219,238),(32,119,220,239),(33,120,221,240),(34,106,222,226),(35,107,223,227),(36,108,224,228),(37,109,225,229),(38,110,211,230),(39,111,212,231),(40,112,213,232),(41,113,214,233),(42,114,215,234),(43,115,216,235),(44,116,217,236),(45,117,218,237),(46,192,84,165),(47,193,85,151),(48,194,86,152),(49,195,87,153),(50,181,88,154),(51,182,89,155),(52,183,90,156),(53,184,76,157),(54,185,77,158),(55,186,78,159),(56,187,79,160),(57,188,80,161),(58,189,81,162),(59,190,82,163),(60,191,83,164)], [(1,45,73,117,145,218,99,237),(2,31,74,118,146,219,100,238),(3,32,75,119,147,220,101,239),(4,33,61,120,148,221,102,240),(5,34,62,106,149,222,103,226),(6,35,63,107,150,223,104,227),(7,36,64,108,136,224,105,228),(8,37,65,109,137,225,91,229),(9,38,66,110,138,211,92,230),(10,39,67,111,139,212,93,231),(11,40,68,112,140,213,94,232),(12,41,69,113,141,214,95,233),(13,42,70,114,142,215,96,234),(14,43,71,115,143,216,97,235),(15,44,72,116,144,217,98,236),(16,152,199,86,127,194,169,48),(17,153,200,87,128,195,170,49),(18,154,201,88,129,181,171,50),(19,155,202,89,130,182,172,51),(20,156,203,90,131,183,173,52),(21,157,204,76,132,184,174,53),(22,158,205,77,133,185,175,54),(23,159,206,78,134,186,176,55),(24,160,207,79,135,187,177,56),(25,161,208,80,121,188,178,57),(26,162,209,81,122,189,179,58),(27,163,210,82,123,190,180,59),(28,164,196,83,124,191,166,60),(29,165,197,84,125,192,167,46),(30,151,198,85,126,193,168,47)], [(1,203),(2,204),(3,205),(4,206),(5,207),(6,208),(7,209),(8,210),(9,196),(10,197),(11,198),(12,199),(13,200),(14,201),(15,202),(16,69),(17,70),(18,71),(19,72),(20,73),(21,74),(22,75),(23,61),(24,62),(25,63),(26,64),(27,65),(28,66),(29,67),(30,68),(31,157),(32,158),(33,159),(34,160),(35,161),(36,162),(37,163),(38,164),(39,165),(40,151),(41,152),(42,153),(43,154),(44,155),(45,156),(46,111),(47,112),(48,113),(49,114),(50,115),(51,116),(52,117),(53,118),(54,119),(55,120),(56,106),(57,107),(58,108),(59,109),(60,110),(76,238),(77,239),(78,240),(79,226),(80,227),(81,228),(82,229),(83,230),(84,231),(85,232),(86,233),(87,234),(88,235),(89,236),(90,237),(91,123),(92,124),(93,125),(94,126),(95,127),(96,128),(97,129),(98,130),(99,131),(100,132),(101,133),(102,134),(103,135),(104,121),(105,122),(136,179),(137,180),(138,166),(139,167),(140,168),(141,169),(142,170),(143,171),(144,172),(145,173),(146,174),(147,175),(148,176),(149,177),(150,178),(181,216),(182,217),(183,218),(184,219),(185,220),(186,221),(187,222),(188,223),(189,224),(190,225),(191,211),(192,212),(193,213),(194,214),(195,215)]])
210 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 15A | ··· | 15H | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | ··· | 20T | 24A | ··· | 24H | 30A | ··· | 30H | 30I | ··· | 30P | 30Q | ··· | 30AF | 40A | ··· | 40P | 60A | ··· | 60P | 60Q | ··· | 60X | 60Y | ··· | 60AN | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 30 | ··· | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | ··· | 60 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
210 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C5 | C6 | C6 | C6 | C6 | C6 | C10 | C10 | C10 | C10 | C10 | C15 | C30 | C30 | C30 | C30 | C30 | D4 | D4 | C3×D4 | C3×D4 | C4○D8 | C5×D4 | C5×D4 | C3×C4○D8 | D4×C15 | D4×C15 | C5×C4○D8 | C15×C4○D8 |
| kernel | C15×C4○D8 | C2×C120 | C15×D8 | C15×SD16 | C15×Q16 | C15×C4○D4 | C5×C4○D8 | C3×C4○D8 | C2×C40 | C5×D8 | C5×SD16 | C5×Q16 | C5×C4○D4 | C2×C24 | C3×D8 | C3×SD16 | C3×Q16 | C3×C4○D4 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C60 | C2×C30 | C20 | C2×C10 | C15 | C12 | C2×C6 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 8 | 8 | 8 | 16 | 8 | 16 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of C15×C4○D8 ►in GL2(𝔽241) generated by
| 100 | 0 |
| 0 | 100 |
| 177 | 0 |
| 0 | 177 |
| 211 | 140 |
| 0 | 8 |
| 183 | 197 |
| 60 | 58 |
G:=sub<GL(2,GF(241))| [100,0,0,100],[177,0,0,177],[211,0,140,8],[183,60,197,58] >;
C15×C4○D8 in GAP, Magma, Sage, TeX
C_{15}\times C_4\circ D_8 % in TeX
G:=Group("C15xC4oD8"); // GroupNames label
G:=SmallGroup(480,940);
// by ID
G=gap.SmallGroup(480,940);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1709,1276,15125,7572,124]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^3>;
// generators/relations