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## G = C15×C4○D8order 480 = 25·3·5

### Direct product of C15 and C4○D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C15×C4○D8
 Chief series C1 — C2 — C4 — C20 — C60 — D4×C15 — C15×D8 — C15×C4○D8
 Lower central C1 — C2 — C4 — C15×C4○D8
 Upper central C1 — C60 — C2×C60 — C15×C4○D8

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 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C5, C6, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], D4 [×2], Q8 [×2], C10, C10 [×3], C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C24 [×2], C2×C12, C2×C12 [×2], C3×D4 [×2], C3×D4 [×2], C3×Q8 [×2], C30, C30 [×3], C4○D8, C40 [×2], C2×C20, C2×C20 [×2], C5×D4 [×2], C5×D4 [×2], C5×Q8 [×2], C2×C24, C3×D8, C3×SD16 [×2], C3×Q16, C3×C4○D4 [×2], C60 [×2], C60 [×2], C2×C30, C2×C30 [×2], C2×C40, C5×D8, C5×SD16 [×2], C5×Q16, C5×C4○D4 [×2], C3×C4○D8, C120 [×2], C2×C60, C2×C60 [×2], D4×C15 [×2], D4×C15 [×2], Q8×C15 [×2], C5×C4○D8, C2×C120, C15×D8, C15×SD16 [×2], C15×Q16, C15×C4○D4 [×2], C15×C4○D8
Quotients: C1, C2 [×7], C3, C22 [×7], C5, C6 [×7], D4 [×2], C23, C10 [×7], C2×C6 [×7], C15, C2×D4, C2×C10 [×7], C3×D4 [×2], C22×C6, C30 [×7], C4○D8, C5×D4 [×2], C22×C10, C6×D4, C2×C30 [×7], D4×C10, C3×C4○D8, D4×C15 [×2], C22×C30, C5×C4○D8, D4×C30, C15×C4○D8

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

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

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

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

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

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