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## G = (C4×D15)⋊8C4order 480 = 25·3·5

### 4th 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)⋊8C4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic3×Dic5 — (C4×D15)⋊8C4
 Lower central C15 — C30 — (C4×D15)⋊8C4
 Upper central C1 — C22 — C2×C4

Generators and relations for (C4×D15)⋊8C4
G = < a,b,c,d | a4=b15=c2=d4=1, ab=ba, ac=ca, dad-1=a-1, cbc=b-1, dbd-1=b11, dcd-1=a2b10c >

Subgroups: 748 in 152 conjugacy classes, 60 normal (32 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×9], C23, D5 [×2], C10 [×3], 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 [×3], C42⋊C2, C4×D5 [×4], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C4×Dic3 [×2], C4⋊Dic3, D6⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], D30 [×2], D30 [×2], C2×C30, C4×Dic5 [×2], C4⋊Dic5, D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C4⋊C47S3, C6×Dic5 [×2], C10×Dic3 [×2], C4×D15 [×4], C2×Dic15, C2×C60, C22×D15, C4⋊C47D5, Dic3×Dic5 [×2], D304C4 [×2], C3×C4⋊Dic5, C5×C4⋊Dic3, C2×C4×D15, (C4×D15)⋊8C4
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, D42S3, Q83S3, S3×D5, C2×C4×D5, D42D5, Q82D5, C4⋊C47S3, D30.C2 [×2], C2×S3×D5, C4⋊C47D5, D20⋊S3, D12⋊D5, C2×D30.C2, (C4×D15)⋊8C4

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

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

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

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

66 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 12F 15A 15B 20A 20B 20C 20D 20E ··· 20L 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 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 30 30 2 2 2 6 6 6 6 10 10 10 10 15 15 15 15 2 2 2 2 2 2 ··· 2 4 4 20 20 20 20 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 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 D4⋊2S3 Q8⋊3S3 S3×D5 D4⋊2D5 Q8⋊2D5 D30.C2 C2×S3×D5 D20⋊S3 D12⋊D5 kernel (C4×D15)⋊8C4 Dic3×Dic5 D30⋊4C4 C3×C4⋊Dic5 C5×C4⋊Dic3 C2×C4×D15 C4×D15 C4⋊Dic5 C4⋊Dic3 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 C20 C12 C10 C10 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 2 1 1 1 8 1 2 2 1 4 4 2 4 8 1 1 2 2 2 4 2 4 4

Matrix representation of (C4×D15)⋊8C4 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 0 0 0 50
,
 59 15 0 0 0 0 12 1 0 0 0 0 0 0 0 18 0 0 0 0 44 17 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 2 46 0 0 0 0 49 59 0 0 0 0 0 0 44 18 0 0 0 0 45 17 0 0 0 0 0 0 60 0 0 0 0 0 0 1
,
 11 0 0 0 0 0 51 50 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,50],[59,12,0,0,0,0,15,1,0,0,0,0,0,0,0,44,0,0,0,0,18,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,49,0,0,0,0,46,59,0,0,0,0,0,0,44,45,0,0,0,0,18,17,0,0,0,0,0,0,60,0,0,0,0,0,0,1],[11,51,0,0,0,0,0,50,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

(C4×D15)⋊8C4 in GAP, Magma, Sage, TeX

(C_4\times D_{15})\rtimes_8C_4
% in TeX

G:=Group("(C4xD15):8C4");
// GroupNames label

G:=SmallGroup(480,423);
// by ID

G=gap.SmallGroup(480,423);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,176,422,219,100,1356,18822]);
// Polycyclic

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

׿
×
𝔽