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## G = C15×C4.4D4order 480 = 25·3·5

### Direct product of C15 and C4.4D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C15×C4.4D4
 Chief series C1 — C2 — C22 — C2×C10 — C2×C30 — C22×C30 — C15×C22⋊C4 — C15×C4.4D4
 Lower central C1 — C22 — C15×C4.4D4
 Upper central C1 — C2×C30 — C15×C4.4D4

Generators and relations for C15×C4.4D4
G = < a,b,c,d | a15=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 232 in 152 conjugacy classes, 88 normal (32 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C5, C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C10, C10 [×2], C10 [×2], C12 [×2], C12 [×4], C2×C6, C2×C6 [×6], C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, C20 [×2], C20 [×4], C2×C10, C2×C10 [×6], C2×C12, C2×C12 [×4], C3×D4 [×2], C3×Q8 [×2], C22×C6 [×2], C30, C30 [×2], C30 [×2], C4.4D4, C2×C20, C2×C20 [×4], C5×D4 [×2], C5×Q8 [×2], C22×C10 [×2], C4×C12, C3×C22⋊C4 [×4], C6×D4, C6×Q8, C60 [×2], C60 [×4], C2×C30, C2×C30 [×6], C4×C20, C5×C22⋊C4 [×4], D4×C10, Q8×C10, C3×C4.4D4, C2×C60, C2×C60 [×4], D4×C15 [×2], Q8×C15 [×2], C22×C30 [×2], C5×C4.4D4, C4×C60, C15×C22⋊C4 [×4], D4×C30, Q8×C30, C15×C4.4D4
Quotients: C1, C2 [×7], C3, C22 [×7], C5, C6 [×7], D4 [×2], C23, C10 [×7], C2×C6 [×7], C15, C2×D4, C4○D4 [×2], C2×C10 [×7], C3×D4 [×2], C22×C6, C30 [×7], C4.4D4, C5×D4 [×2], C22×C10, C6×D4, C3×C4○D4 [×2], C2×C30 [×7], D4×C10, C5×C4○D4 [×2], C3×C4.4D4, D4×C15 [×2], C22×C30, C5×C4.4D4, D4×C30, C15×C4○D4 [×2], C15×C4.4D4

Smallest permutation representation of C15×C4.4D4
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 76 165 227)(2 77 151 228)(3 78 152 229)(4 79 153 230)(5 80 154 231)(6 81 155 232)(7 82 156 233)(8 83 157 234)(9 84 158 235)(10 85 159 236)(11 86 160 237)(12 87 161 238)(13 88 162 239)(14 89 163 240)(15 90 164 226)(16 52 121 212)(17 53 122 213)(18 54 123 214)(19 55 124 215)(20 56 125 216)(21 57 126 217)(22 58 127 218)(23 59 128 219)(24 60 129 220)(25 46 130 221)(26 47 131 222)(27 48 132 223)(28 49 133 224)(29 50 134 225)(30 51 135 211)(31 196 188 72)(32 197 189 73)(33 198 190 74)(34 199 191 75)(35 200 192 61)(36 201 193 62)(37 202 194 63)(38 203 195 64)(39 204 181 65)(40 205 182 66)(41 206 183 67)(42 207 184 68)(43 208 185 69)(44 209 186 70)(45 210 187 71)(91 116 137 169)(92 117 138 170)(93 118 139 171)(94 119 140 172)(95 120 141 173)(96 106 142 174)(97 107 143 175)(98 108 144 176)(99 109 145 177)(100 110 146 178)(101 111 147 179)(102 112 148 180)(103 113 149 166)(104 114 150 167)(105 115 136 168)
(1 184 178 218)(2 185 179 219)(3 186 180 220)(4 187 166 221)(5 188 167 222)(6 189 168 223)(7 190 169 224)(8 191 170 225)(9 192 171 211)(10 193 172 212)(11 194 173 213)(12 195 174 214)(13 181 175 215)(14 182 176 216)(15 183 177 217)(16 85 62 94)(17 86 63 95)(18 87 64 96)(19 88 65 97)(20 89 66 98)(21 90 67 99)(22 76 68 100)(23 77 69 101)(24 78 70 102)(25 79 71 103)(26 80 72 104)(27 81 73 105)(28 82 74 91)(29 83 75 92)(30 84 61 93)(31 114 47 154)(32 115 48 155)(33 116 49 156)(34 117 50 157)(35 118 51 158)(36 119 52 159)(37 120 53 160)(38 106 54 161)(39 107 55 162)(40 108 56 163)(41 109 57 164)(42 110 58 165)(43 111 59 151)(44 112 60 152)(45 113 46 153)(121 236 201 140)(122 237 202 141)(123 238 203 142)(124 239 204 143)(125 240 205 144)(126 226 206 145)(127 227 207 146)(128 228 208 147)(129 229 209 148)(130 230 210 149)(131 231 196 150)(132 232 197 136)(133 233 198 137)(134 234 199 138)(135 235 200 139)
(1 22 165 127)(2 23 151 128)(3 24 152 129)(4 25 153 130)(5 26 154 131)(6 27 155 132)(7 28 156 133)(8 29 157 134)(9 30 158 135)(10 16 159 121)(11 17 160 122)(12 18 161 123)(13 19 162 124)(14 20 163 125)(15 21 164 126)(31 104 188 150)(32 105 189 136)(33 91 190 137)(34 92 191 138)(35 93 192 139)(36 94 193 140)(37 95 194 141)(38 96 195 142)(39 97 181 143)(40 98 182 144)(41 99 183 145)(42 100 184 146)(43 101 185 147)(44 102 186 148)(45 103 187 149)(46 79 221 230)(47 80 222 231)(48 81 223 232)(49 82 224 233)(50 83 225 234)(51 84 211 235)(52 85 212 236)(53 86 213 237)(54 87 214 238)(55 88 215 239)(56 89 216 240)(57 90 217 226)(58 76 218 227)(59 77 219 228)(60 78 220 229)(61 118 200 171)(62 119 201 172)(63 120 202 173)(64 106 203 174)(65 107 204 175)(66 108 205 176)(67 109 206 177)(68 110 207 178)(69 111 208 179)(70 112 209 180)(71 113 210 166)(72 114 196 167)(73 115 197 168)(74 116 198 169)(75 117 199 170)

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

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

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

210 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A ··· 4F 4G 4H 5A 5B 5C 5D 6A ··· 6F 6G 6H 6I 6J 10A ··· 10L 10M ··· 10T 12A ··· 12L 12M 12N 12O 12P 15A ··· 15H 20A ··· 20X 20Y ··· 20AF 30A ··· 30X 30Y ··· 30AN 60A ··· 60AV 60AW ··· 60BL order 1 2 2 2 2 2 3 3 4 ··· 4 4 4 5 5 5 5 6 ··· 6 6 6 6 6 10 ··· 10 10 ··· 10 12 ··· 12 12 12 12 12 15 ··· 15 20 ··· 20 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 60 ··· 60 size 1 1 1 1 4 4 1 1 2 ··· 2 4 4 1 1 1 1 1 ··· 1 4 4 4 4 1 ··· 1 4 ··· 4 2 ··· 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 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 type + + + + + + image C1 C2 C2 C2 C2 C3 C5 C6 C6 C6 C6 C10 C10 C10 C10 C15 C30 C30 C30 C30 D4 C4○D4 C3×D4 C5×D4 C3×C4○D4 C5×C4○D4 D4×C15 C15×C4○D4 kernel C15×C4.4D4 C4×C60 C15×C22⋊C4 D4×C30 Q8×C30 C5×C4.4D4 C3×C4.4D4 C4×C20 C5×C22⋊C4 D4×C10 Q8×C10 C4×C12 C3×C22⋊C4 C6×D4 C6×Q8 C4.4D4 C42 C22⋊C4 C2×D4 C2×Q8 C60 C30 C20 C12 C10 C6 C4 C2 # reps 1 1 4 1 1 2 4 2 8 2 2 4 16 4 4 8 8 32 8 8 2 4 4 8 8 16 16 32

Matrix representation of C15×C4.4D4 in GL4(𝔽61) generated by

 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 37 32 0 0 22 24 0 0 0 0 10 22 0 0 37 51
,
 41 47 0 0 59 20 0 0 0 0 11 0 0 0 0 11
,
 50 37 0 0 0 11 0 0 0 0 49 59 0 0 42 12
G:=sub<GL(4,GF(61))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[37,22,0,0,32,24,0,0,0,0,10,37,0,0,22,51],[41,59,0,0,47,20,0,0,0,0,11,0,0,0,0,11],[50,0,0,0,37,11,0,0,0,0,49,42,0,0,59,12] >;

C15×C4.4D4 in GAP, Magma, Sage, TeX

C_{15}\times C_4._4D_4
% in TeX

G:=Group("C15xC4.4D4");
// GroupNames label

G:=SmallGroup(480,929);
// by ID

G=gap.SmallGroup(480,929);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1709,1688,5126,646]);
// Polycyclic

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

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