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## G = C15×C4⋊D4order 480 = 25·3·5

### Direct product of C15 and C4⋊D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C15×C4⋊D4
 Chief series C1 — C2 — C22 — C2×C10 — C2×C30 — C22×C30 — D4×C30 — C15×C4⋊D4
 Lower central C1 — C22 — C15×C4⋊D4
 Upper central C1 — C2×C30 — C15×C4⋊D4

Generators and relations for C15×C4⋊D4
G = < a,b,c,d | a15=b4=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 296 in 188 conjugacy classes, 96 normal (48 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], C5, C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×2], D4 [×6], C23, C23 [×2], C10 [×3], C10 [×4], C12 [×2], C12 [×3], C2×C6, C2×C6 [×2], C2×C6 [×8], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], C20 [×2], C20 [×3], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C12 [×2], C2×C12 [×2], C2×C12 [×2], C3×D4 [×6], C22×C6, C22×C6 [×2], C30 [×3], C30 [×4], C4⋊D4, C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×6], C22×C10, C22×C10 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4, C22×C12, C6×D4, C6×D4 [×2], C60 [×2], C60 [×3], C2×C30, C2×C30 [×2], C2×C30 [×8], C5×C22⋊C4 [×2], C5×C4⋊C4, C22×C20, D4×C10, D4×C10 [×2], C3×C4⋊D4, C2×C60 [×2], C2×C60 [×2], C2×C60 [×2], D4×C15 [×6], C22×C30, C22×C30 [×2], C5×C4⋊D4, C15×C22⋊C4 [×2], C15×C4⋊C4, C22×C60, D4×C30, D4×C30 [×2], C15×C4⋊D4
Quotients: C1, C2 [×7], C3, C22 [×7], C5, C6 [×7], D4 [×4], C23, C10 [×7], C2×C6 [×7], C15, C2×D4 [×2], C4○D4, C2×C10 [×7], C3×D4 [×4], C22×C6, C30 [×7], C4⋊D4, C5×D4 [×4], C22×C10, C6×D4 [×2], C3×C4○D4, C2×C30 [×7], D4×C10 [×2], C5×C4○D4, C3×C4⋊D4, D4×C15 [×4], C22×C30, C5×C4⋊D4, D4×C30 [×2], C15×C4○D4, C15×C4⋊D4

Smallest permutation representation of C15×C4⋊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)
(1 137 19 126)(2 138 20 127)(3 139 21 128)(4 140 22 129)(5 141 23 130)(6 142 24 131)(7 143 25 132)(8 144 26 133)(9 145 27 134)(10 146 28 135)(11 147 29 121)(12 148 30 122)(13 149 16 123)(14 150 17 124)(15 136 18 125)(31 97 201 233)(32 98 202 234)(33 99 203 235)(34 100 204 236)(35 101 205 237)(36 102 206 238)(37 103 207 239)(38 104 208 240)(39 105 209 226)(40 91 210 227)(41 92 196 228)(42 93 197 229)(43 94 198 230)(44 95 199 231)(45 96 200 232)(46 162 118 180)(47 163 119 166)(48 164 120 167)(49 165 106 168)(50 151 107 169)(51 152 108 170)(52 153 109 171)(53 154 110 172)(54 155 111 173)(55 156 112 174)(56 157 113 175)(57 158 114 176)(58 159 115 177)(59 160 116 178)(60 161 117 179)(61 90 186 219)(62 76 187 220)(63 77 188 221)(64 78 189 222)(65 79 190 223)(66 80 191 224)(67 81 192 225)(68 82 193 211)(69 83 194 212)(70 84 195 213)(71 85 181 214)(72 86 182 215)(73 87 183 216)(74 88 184 217)(75 89 185 218)
(1 87 197 159)(2 88 198 160)(3 89 199 161)(4 90 200 162)(5 76 201 163)(6 77 202 164)(7 78 203 165)(8 79 204 151)(9 80 205 152)(10 81 206 153)(11 82 207 154)(12 83 208 155)(13 84 209 156)(14 85 210 157)(15 86 196 158)(16 213 39 174)(17 214 40 175)(18 215 41 176)(19 216 42 177)(20 217 43 178)(21 218 44 179)(22 219 45 180)(23 220 31 166)(24 221 32 167)(25 222 33 168)(26 223 34 169)(27 224 35 170)(28 225 36 171)(29 211 37 172)(30 212 38 173)(46 140 61 232)(47 141 62 233)(48 142 63 234)(49 143 64 235)(50 144 65 236)(51 145 66 237)(52 146 67 238)(53 147 68 239)(54 148 69 240)(55 149 70 226)(56 150 71 227)(57 136 72 228)(58 137 73 229)(59 138 74 230)(60 139 75 231)(91 113 124 181)(92 114 125 182)(93 115 126 183)(94 116 127 184)(95 117 128 185)(96 118 129 186)(97 119 130 187)(98 120 131 188)(99 106 132 189)(100 107 133 190)(101 108 134 191)(102 109 135 192)(103 110 121 193)(104 111 122 194)(105 112 123 195)
(1 159)(2 160)(3 161)(4 162)(5 163)(6 164)(7 165)(8 151)(9 152)(10 153)(11 154)(12 155)(13 156)(14 157)(15 158)(16 174)(17 175)(18 176)(19 177)(20 178)(21 179)(22 180)(23 166)(24 167)(25 168)(26 169)(27 170)(28 171)(29 172)(30 173)(31 220)(32 221)(33 222)(34 223)(35 224)(36 225)(37 211)(38 212)(39 213)(40 214)(41 215)(42 216)(43 217)(44 218)(45 219)(46 140)(47 141)(48 142)(49 143)(50 144)(51 145)(52 146)(53 147)(54 148)(55 149)(56 150)(57 136)(58 137)(59 138)(60 139)(61 232)(62 233)(63 234)(64 235)(65 236)(66 237)(67 238)(68 239)(69 240)(70 226)(71 227)(72 228)(73 229)(74 230)(75 231)(76 201)(77 202)(78 203)(79 204)(80 205)(81 206)(82 207)(83 208)(84 209)(85 210)(86 196)(87 197)(88 198)(89 199)(90 200)(91 181)(92 182)(93 183)(94 184)(95 185)(96 186)(97 187)(98 188)(99 189)(100 190)(101 191)(102 192)(103 193)(104 194)(105 195)(106 132)(107 133)(108 134)(109 135)(110 121)(111 122)(112 123)(113 124)(114 125)(115 126)(116 127)(117 128)(118 129)(119 130)(120 131)

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

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

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

210 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 6N 10A ··· 10L 10M ··· 10T 10U ··· 10AB 12A ··· 12H 12I 12J 12K 12L 15A ··· 15H 20A ··· 20P 20Q ··· 20X 30A ··· 30X 30Y ··· 30AN 30AO ··· 30BD 60A ··· 60AF 60AG ··· 60AV order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 5 5 5 5 6 ··· 6 6 6 6 6 6 6 6 6 10 ··· 10 10 ··· 10 10 ··· 10 12 ··· 12 12 12 12 12 15 ··· 15 20 ··· 20 20 ··· 20 30 ··· 30 30 ··· 30 30 ··· 30 60 ··· 60 60 ··· 60 size 1 1 1 1 2 2 4 4 1 1 2 2 2 2 4 4 1 1 1 1 1 ··· 1 2 2 2 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

210 irreducible representations

 dim 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 C3 C5 C6 C6 C6 C6 C10 C10 C10 C10 C15 C30 C30 C30 C30 D4 D4 C4○D4 C3×D4 C3×D4 C5×D4 C5×D4 C3×C4○D4 C5×C4○D4 D4×C15 D4×C15 C15×C4○D4 kernel C15×C4⋊D4 C15×C22⋊C4 C15×C4⋊C4 C22×C60 D4×C30 C5×C4⋊D4 C3×C4⋊D4 C5×C22⋊C4 C5×C4⋊C4 C22×C20 D4×C10 C3×C22⋊C4 C3×C4⋊C4 C22×C12 C6×D4 C4⋊D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C60 C2×C30 C30 C20 C2×C10 C12 C2×C6 C10 C6 C4 C22 C2 # reps 1 2 1 1 3 2 4 4 2 2 6 8 4 4 12 8 16 8 8 24 2 2 2 4 4 8 8 4 8 16 16 16

Matrix representation of C15×C4⋊D4 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 25 0 0 0 0 25
,
 11 0 0 0 8 50 0 0 0 0 1 0 0 0 0 1
,
 34 59 0 0 60 27 0 0 0 0 17 1 0 0 15 44
,
 34 59 0 0 59 27 0 0 0 0 17 1 0 0 17 44
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,25,0,0,0,0,25],[11,8,0,0,0,50,0,0,0,0,1,0,0,0,0,1],[34,60,0,0,59,27,0,0,0,0,17,15,0,0,1,44],[34,59,0,0,59,27,0,0,0,0,17,17,0,0,1,44] >;

C15×C4⋊D4 in GAP, Magma, Sage, TeX

C_{15}\times C_4\rtimes D_4
% in TeX

G:=Group("C15xC4:D4");
// GroupNames label

G:=SmallGroup(480,926);
// by ID

G=gap.SmallGroup(480,926);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1709,848,5126]);
// Polycyclic

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

׿
×
𝔽