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## G = C2×D30⋊3C4order 480 = 25·3·5

### Direct product of C2 and D30⋊3C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×D30⋊3C4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C22×D15 — C23×D15 — C2×D30⋊3C4
 Lower central C15 — C30 — C2×D30⋊3C4
 Upper central C1 — C23 — C22×C4

Generators and relations for C2×D303C4
G = < a,b,c,d | a2=b30=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b15c >

Subgroups: 1716 in 264 conjugacy classes, 95 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, C23, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, D15, C30, C30, C2×C22⋊C4, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×C10, D6⋊C4, C22×Dic3, C22×C12, S3×C23, Dic15, C60, D30, D30, C2×C30, C2×C30, D10⋊C4, C22×Dic5, C22×C20, C23×D5, C2×D6⋊C4, C2×Dic15, C2×Dic15, C2×C60, C2×C60, C22×D15, C22×D15, C22×C30, C2×D10⋊C4, D303C4, C22×Dic15, C22×C60, C23×D15, C2×D303C4
Quotients:

Smallest permutation representation of C2×D303C4
On 240 points
Generators in S240
(1 152)(2 153)(3 154)(4 155)(5 156)(6 157)(7 158)(8 159)(9 160)(10 161)(11 162)(12 163)(13 164)(14 165)(15 166)(16 167)(17 168)(18 169)(19 170)(20 171)(21 172)(22 173)(23 174)(24 175)(25 176)(26 177)(27 178)(28 179)(29 180)(30 151)(31 127)(32 128)(33 129)(34 130)(35 131)(36 132)(37 133)(38 134)(39 135)(40 136)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 145)(50 146)(51 147)(52 148)(53 149)(54 150)(55 121)(56 122)(57 123)(58 124)(59 125)(60 126)(61 240)(62 211)(63 212)(64 213)(65 214)(66 215)(67 216)(68 217)(69 218)(70 219)(71 220)(72 221)(73 222)(74 223)(75 224)(76 225)(77 226)(78 227)(79 228)(80 229)(81 230)(82 231)(83 232)(84 233)(85 234)(86 235)(87 236)(88 237)(89 238)(90 239)(91 205)(92 206)(93 207)(94 208)(95 209)(96 210)(97 181)(98 182)(99 183)(100 184)(101 185)(102 186)(103 187)(104 188)(105 189)(106 190)(107 191)(108 192)(109 193)(110 194)(111 195)(112 196)(113 197)(114 198)(115 199)(116 200)(117 201)(118 202)(119 203)(120 204)
(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)
(1 166)(2 165)(3 164)(4 163)(5 162)(6 161)(7 160)(8 159)(9 158)(10 157)(11 156)(12 155)(13 154)(14 153)(15 152)(16 151)(17 180)(18 179)(19 178)(20 177)(21 176)(22 175)(23 174)(24 173)(25 172)(26 171)(27 170)(28 169)(29 168)(30 167)(31 149)(32 148)(33 147)(34 146)(35 145)(36 144)(37 143)(38 142)(39 141)(40 140)(41 139)(42 138)(43 137)(44 136)(45 135)(46 134)(47 133)(48 132)(49 131)(50 130)(51 129)(52 128)(53 127)(54 126)(55 125)(56 124)(57 123)(58 122)(59 121)(60 150)(61 237)(62 236)(63 235)(64 234)(65 233)(66 232)(67 231)(68 230)(69 229)(70 228)(71 227)(72 226)(73 225)(74 224)(75 223)(76 222)(77 221)(78 220)(79 219)(80 218)(81 217)(82 216)(83 215)(84 214)(85 213)(86 212)(87 211)(88 240)(89 239)(90 238)(91 206)(92 205)(93 204)(94 203)(95 202)(96 201)(97 200)(98 199)(99 198)(100 197)(101 196)(102 195)(103 194)(104 193)(105 192)(106 191)(107 190)(108 189)(109 188)(110 187)(111 186)(112 185)(113 184)(114 183)(115 182)(116 181)(117 210)(118 209)(119 208)(120 207)
(1 107 35 75)(2 108 36 76)(3 109 37 77)(4 110 38 78)(5 111 39 79)(6 112 40 80)(7 113 41 81)(8 114 42 82)(9 115 43 83)(10 116 44 84)(11 117 45 85)(12 118 46 86)(13 119 47 87)(14 120 48 88)(15 91 49 89)(16 92 50 90)(17 93 51 61)(18 94 52 62)(19 95 53 63)(20 96 54 64)(21 97 55 65)(22 98 56 66)(23 99 57 67)(24 100 58 68)(25 101 59 69)(26 102 60 70)(27 103 31 71)(28 104 32 72)(29 105 33 73)(30 106 34 74)(121 214 172 181)(122 215 173 182)(123 216 174 183)(124 217 175 184)(125 218 176 185)(126 219 177 186)(127 220 178 187)(128 221 179 188)(129 222 180 189)(130 223 151 190)(131 224 152 191)(132 225 153 192)(133 226 154 193)(134 227 155 194)(135 228 156 195)(136 229 157 196)(137 230 158 197)(138 231 159 198)(139 232 160 199)(140 233 161 200)(141 234 162 201)(142 235 163 202)(143 236 164 203)(144 237 165 204)(145 238 166 205)(146 239 167 206)(147 240 168 207)(148 211 169 208)(149 212 170 209)(150 213 171 210)

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

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

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

132 conjugacy classes

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

132 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D4 D5 D6 D6 D10 D10 C4×S3 D12 C3⋊D4 D15 C4×D5 D20 C5⋊D4 D30 D30 C4×D15 D60 C15⋊7D4 kernel C2×D30⋊3C4 D30⋊3C4 C22×Dic15 C22×C60 C23×D15 C22×D15 C22×C20 C2×C30 C22×C12 C2×C20 C22×C10 C2×C12 C22×C6 C2×C10 C2×C10 C2×C10 C22×C4 C2×C6 C2×C6 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 4 1 1 1 8 1 4 2 2 1 4 2 4 4 4 4 8 8 8 8 4 16 16 16

Matrix representation of C2×D303C4 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 60 0 0 0 0 0 0 60
,
 60 60 0 0 0 0 1 0 0 0 0 0 0 0 37 8 0 0 0 0 6 31 0 0 0 0 0 0 36 9 0 0 0 0 22 14
,
 1 1 0 0 0 0 0 60 0 0 0 0 0 0 25 45 0 0 0 0 39 36 0 0 0 0 0 0 37 16 0 0 0 0 6 24
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 32 4 0 0 0 0 3 29

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,1,0,0,0,0,60,0,0,0,0,0,0,0,37,6,0,0,0,0,8,31,0,0,0,0,0,0,36,22,0,0,0,0,9,14],[1,0,0,0,0,0,1,60,0,0,0,0,0,0,25,39,0,0,0,0,45,36,0,0,0,0,0,0,37,6,0,0,0,0,16,24],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,32,3,0,0,0,0,4,29] >;

C2×D303C4 in GAP, Magma, Sage, TeX

C_2\times D_{30}\rtimes_3C_4
% in TeX

G:=Group("C2xD30:3C4");
// GroupNames label

G:=SmallGroup(480,892);
// by ID

G=gap.SmallGroup(480,892);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,58,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^30=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^15*c>;
// generators/relations

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