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## G = C15×C8○D4order 480 = 25·3·5

### Direct product of C15 and C8○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C15×C8○D4
 Chief series C1 — C2 — C4 — C20 — C60 — C120 — C2×C120 — C15×C8○D4
 Lower central C1 — C2 — C15×C8○D4
 Upper central C1 — C120 — C15×C8○D4

Generators and relations for C15×C8○D4
G = < a,b,c,d | a15=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >

Subgroups: 136 in 124 conjugacy classes, 112 normal (28 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, C6, C6 [×3], C8, C8 [×3], C2×C4 [×3], D4 [×3], Q8, C10, C10 [×3], C12, C12 [×3], C2×C6 [×3], C15, C2×C8 [×3], M4(2) [×3], C4○D4, C20, C20 [×3], C2×C10 [×3], C24, C24 [×3], C2×C12 [×3], C3×D4 [×3], C3×Q8, C30, C30 [×3], C8○D4, C40, C40 [×3], C2×C20 [×3], C5×D4 [×3], C5×Q8, C2×C24 [×3], C3×M4(2) [×3], C3×C4○D4, C60, C60 [×3], C2×C30 [×3], C2×C40 [×3], C5×M4(2) [×3], C5×C4○D4, C3×C8○D4, C120, C120 [×3], C2×C60 [×3], D4×C15 [×3], Q8×C15, C5×C8○D4, C2×C120 [×3], C15×M4(2) [×3], C15×C4○D4, C15×C8○D4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C5, C6 [×7], C2×C4 [×6], C23, C10 [×7], C12 [×4], C2×C6 [×7], C15, C22×C4, C20 [×4], C2×C10 [×7], C2×C12 [×6], C22×C6, C30 [×7], C8○D4, C2×C20 [×6], C22×C10, C22×C12, C60 [×4], C2×C30 [×7], C22×C20, C3×C8○D4, C2×C60 [×6], C22×C30, C5×C8○D4, C22×C60, C15×C8○D4

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

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

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

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

300 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 5A 5B 5C 5D 6A 6B 6C ··· 6H 8A 8B 8C 8D 8E ··· 8J 10A 10B 10C 10D 10E ··· 10P 12A 12B 12C 12D 12E ··· 12J 15A ··· 15H 20A ··· 20H 20I ··· 20T 24A ··· 24H 24I ··· 24T 30A ··· 30H 30I ··· 30AF 40A ··· 40P 40Q ··· 40AN 60A ··· 60P 60Q ··· 60AN 120A ··· 120AF 120AG ··· 120CB order 1 2 2 2 2 3 3 4 4 4 4 4 5 5 5 5 6 6 6 ··· 6 8 8 8 8 8 ··· 8 10 10 10 10 10 ··· 10 12 12 12 12 12 ··· 12 15 ··· 15 20 ··· 20 20 ··· 20 24 ··· 24 24 ··· 24 30 ··· 30 30 ··· 30 40 ··· 40 40 ··· 40 60 ··· 60 60 ··· 60 120 ··· 120 120 ··· 120 size 1 1 2 2 2 1 1 1 1 2 2 2 1 1 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

300 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 type + + + + image C1 C2 C2 C2 C3 C4 C4 C5 C6 C6 C6 C10 C10 C10 C12 C12 C15 C20 C20 C30 C30 C30 C60 C60 C8○D4 C3×C8○D4 C5×C8○D4 C15×C8○D4 kernel C15×C8○D4 C2×C120 C15×M4(2) C15×C4○D4 C5×C8○D4 D4×C15 Q8×C15 C3×C8○D4 C2×C40 C5×M4(2) C5×C4○D4 C2×C24 C3×M4(2) C3×C4○D4 C5×D4 C5×Q8 C8○D4 C3×D4 C3×Q8 C2×C8 M4(2) C4○D4 D4 Q8 C15 C5 C3 C1 # reps 1 3 3 1 2 6 2 4 6 6 2 12 12 4 12 4 8 24 8 24 24 8 48 16 4 8 16 32

Matrix representation of C15×C8○D4 in GL3(𝔽241) generated by

 98 0 0 0 15 0 0 0 15
,
 240 0 0 0 30 0 0 0 30
,
 240 0 0 0 171 2 0 80 70
,
 240 0 0 0 1 0 0 70 240
G:=sub<GL(3,GF(241))| [98,0,0,0,15,0,0,0,15],[240,0,0,0,30,0,0,0,30],[240,0,0,0,171,80,0,2,70],[240,0,0,0,1,70,0,0,240] >;

C15×C8○D4 in GAP, Magma, Sage, TeX

C_{15}\times C_8\circ D_4
% in TeX

G:=Group("C15xC8oD4");
// GroupNames label

G:=SmallGroup(480,936);
// by ID

G=gap.SmallGroup(480,936);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,840,2571,124]);
// Polycyclic

G:=Group<a,b,c,d|a^15=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations

׿
×
𝔽