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## G = (C4×D15)⋊10C4order 480 = 25·3·5

### 6th semidirect product of C4×D15 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — (C4×D15)⋊10C4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic15⋊5C4 — (C4×D15)⋊10C4
 Lower central C15 — C30 — (C4×D15)⋊10C4
 Upper central C1 — C2×C4

Generators and relations for (C4×D15)⋊10C4
G = < a,b,c,d | a4=b15=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b11, dcd-1=a2b10c >

Subgroups: 748 in 152 conjugacy classes, 60 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×4], C5, S3 [×2], C6, C6 [×2], C2×C4, C2×C4 [×9], C23, D5 [×2], C10, C10 [×2], Dic3 [×4], C12 [×2], C12 [×2], D6 [×4], C2×C6, C15, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, Dic5 [×4], C20 [×2], C20 [×2], D10 [×4], C2×C10, C4×S3 [×4], C2×Dic3 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, D15 [×2], C30, C30 [×2], C42⋊C2, C4×D5 [×4], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C4×Dic3, Dic3⋊C4 [×2], D6⋊C4 [×2], C4×C12, S3×C2×C4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], D30 [×2], D30 [×2], C2×C30, C4×Dic5, C10.D4 [×2], D10⋊C4 [×2], C4×C20, C2×C4×D5, C422S3, C6×Dic5 [×2], C10×Dic3 [×2], C4×D15 [×4], C2×Dic15, C2×C60, C22×D15, C42⋊D5, D304C4 [×2], Dic155C4 [×2], C12×Dic5, Dic3×C20, C2×C4×D15, (C4×D15)⋊10C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], C22×C4, C4○D4 [×2], D10 [×3], C4×S3 [×2], C22×S3, C42⋊C2, C4×D5 [×2], C22×D5, S3×C2×C4, C4○D12 [×2], S3×D5, C2×C4×D5, C4○D20 [×2], C422S3, D30.C2 [×2], C2×S3×D5, C42⋊D5, D6.D10 [×2], C2×D30.C2, (C4×D15)⋊10C4

Smallest permutation representation of (C4×D15)⋊10C4
On 240 points
Generators in S240
(1 85 24 70)(2 86 25 71)(3 87 26 72)(4 88 27 73)(5 89 28 74)(6 90 29 75)(7 76 30 61)(8 77 16 62)(9 78 17 63)(10 79 18 64)(11 80 19 65)(12 81 20 66)(13 82 21 67)(14 83 22 68)(15 84 23 69)(31 113 54 94)(32 114 55 95)(33 115 56 96)(34 116 57 97)(35 117 58 98)(36 118 59 99)(37 119 60 100)(38 120 46 101)(39 106 47 102)(40 107 48 103)(41 108 49 104)(42 109 50 105)(43 110 51 91)(44 111 52 92)(45 112 53 93)(121 200 147 181)(122 201 148 182)(123 202 149 183)(124 203 150 184)(125 204 136 185)(126 205 137 186)(127 206 138 187)(128 207 139 188)(129 208 140 189)(130 209 141 190)(131 210 142 191)(132 196 143 192)(133 197 144 193)(134 198 145 194)(135 199 146 195)(151 235 175 218)(152 236 176 219)(153 237 177 220)(154 238 178 221)(155 239 179 222)(156 240 180 223)(157 226 166 224)(158 227 167 225)(159 228 168 211)(160 229 169 212)(161 230 170 213)(162 231 171 214)(163 232 172 215)(164 233 173 216)(165 234 174 217)
(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 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 36)(10 35)(11 34)(12 33)(13 32)(14 31)(15 45)(16 60)(17 59)(18 58)(19 57)(20 56)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)(29 47)(30 46)(61 101)(62 100)(63 99)(64 98)(65 97)(66 96)(67 95)(68 94)(69 93)(70 92)(71 91)(72 105)(73 104)(74 103)(75 102)(76 120)(77 119)(78 118)(79 117)(80 116)(81 115)(82 114)(83 113)(84 112)(85 111)(86 110)(87 109)(88 108)(89 107)(90 106)(121 170)(122 169)(123 168)(124 167)(125 166)(126 180)(127 179)(128 178)(129 177)(130 176)(131 175)(132 174)(133 173)(134 172)(135 171)(136 157)(137 156)(138 155)(139 154)(140 153)(141 152)(142 151)(143 165)(144 164)(145 163)(146 162)(147 161)(148 160)(149 159)(150 158)(181 230)(182 229)(183 228)(184 227)(185 226)(186 240)(187 239)(188 238)(189 237)(190 236)(191 235)(192 234)(193 233)(194 232)(195 231)(196 217)(197 216)(198 215)(199 214)(200 213)(201 212)(202 211)(203 225)(204 224)(205 223)(206 222)(207 221)(208 220)(209 219)(210 218)
(1 151 45 132)(2 162 31 128)(3 158 32 124)(4 154 33 135)(5 165 34 131)(6 161 35 127)(7 157 36 123)(8 153 37 134)(9 164 38 130)(10 160 39 126)(11 156 40 122)(12 152 41 133)(13 163 42 129)(14 159 43 125)(15 155 44 121)(16 177 60 145)(17 173 46 141)(18 169 47 137)(19 180 48 148)(20 176 49 144)(21 172 50 140)(22 168 51 136)(23 179 52 147)(24 175 53 143)(25 171 54 139)(26 167 55 150)(27 178 56 146)(28 174 57 142)(29 170 58 138)(30 166 59 149)(61 224 99 183)(62 220 100 194)(63 216 101 190)(64 212 102 186)(65 223 103 182)(66 219 104 193)(67 215 105 189)(68 211 91 185)(69 222 92 181)(70 218 93 192)(71 214 94 188)(72 225 95 184)(73 221 96 195)(74 217 97 191)(75 213 98 187)(76 226 118 202)(77 237 119 198)(78 233 120 209)(79 229 106 205)(80 240 107 201)(81 236 108 197)(82 232 109 208)(83 228 110 204)(84 239 111 200)(85 235 112 196)(86 231 113 207)(87 227 114 203)(88 238 115 199)(89 234 116 210)(90 230 117 206)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 5A 5B 6A 6B 6C 10A ··· 10F 12A 12B 12C 12D 12E ··· 12L 15A 15B 20A ··· 20H 20I ··· 20X 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 10 ··· 10 12 12 12 12 12 ··· 12 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 30 30 2 1 1 1 1 6 6 6 6 10 10 10 10 30 30 2 2 2 2 2 2 ··· 2 2 2 2 2 10 ··· 10 4 4 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 S3 D5 D6 D6 C4○D4 D10 D10 C4×S3 C4×D5 C4○D12 C4○D20 S3×D5 D30.C2 C2×S3×D5 D6.D10 kernel (C4×D15)⋊10C4 D30⋊4C4 Dic15⋊5C4 C12×Dic5 Dic3×C20 C2×C4×D15 C4×D15 C4×Dic5 C4×Dic3 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 C20 C12 C10 C6 C2×C4 C4 C22 C2 # reps 1 2 2 1 1 1 8 1 2 2 1 4 4 2 4 8 8 16 2 4 2 8

Matrix representation of (C4×D15)⋊10C4 in GL4(𝔽61) generated by

 50 0 0 0 0 50 0 0 0 0 50 0 0 0 0 50
,
 18 18 0 0 43 60 0 0 0 0 1 41 0 0 52 59
,
 18 18 0 0 60 43 0 0 0 0 59 20 0 0 9 2
,
 31 17 0 0 44 30 0 0 0 0 34 58 0 0 40 27
G:=sub<GL(4,GF(61))| [50,0,0,0,0,50,0,0,0,0,50,0,0,0,0,50],[18,43,0,0,18,60,0,0,0,0,1,52,0,0,41,59],[18,60,0,0,18,43,0,0,0,0,59,9,0,0,20,2],[31,44,0,0,17,30,0,0,0,0,34,40,0,0,58,27] >;

(C4×D15)⋊10C4 in GAP, Magma, Sage, TeX

(C_4\times D_{15})\rtimes_{10}C_4
% in TeX

G:=Group("(C4xD15):10C4");
// GroupNames label

G:=SmallGroup(480,462);
// by ID

G=gap.SmallGroup(480,462);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,64,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^15=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^11,d*c*d^-1=a^2*b^10*c>;
// generators/relations

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