direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C15×C8.C22, Q16⋊2C30, SD16⋊2C30, C60.249D4, M4(2)⋊2C30, C120.76C22, C60.297C23, C8.(C2×C30), (C2×Q8)⋊6C30, (C5×Q16)⋊6C6, C40.13(C2×C6), (Q8×C10)⋊15C6, (C6×Q8)⋊11C10, (Q8×C30)⋊25C2, (C3×Q16)⋊6C10, (C5×SD16)⋊6C6, C4○D4.4C30, D4.3(C2×C30), C12.64(C5×D4), C6.79(D4×C10), C4.15(D4×C15), C10.79(C6×D4), C2.16(D4×C30), C20.64(C3×D4), Q8.6(C2×C30), C24.12(C2×C10), (C15×Q16)⋊14C2, (C3×SD16)⋊6C10, (C2×C30).132D4, C30.462(C2×D4), (C5×M4(2))⋊6C6, C4.6(C22×C30), C22.6(D4×C15), (C15×SD16)⋊14C2, (C3×M4(2))⋊4C10, C20.49(C22×C6), (C15×M4(2))⋊12C2, C12.49(C22×C10), (C2×C60).441C22, (D4×C15).52C22, (Q8×C15).57C22, (C2×C4).8(C2×C30), (C5×C4○D4).9C6, (C2×C6).25(C5×D4), (C2×C20).70(C2×C6), (C3×C4○D4).5C10, (C5×D4).13(C2×C6), (C2×C10).26(C3×D4), (C5×Q8).22(C2×C6), (C2×C12).69(C2×C10), (C15×C4○D4).11C2, (C3×D4).13(C2×C10), (C3×Q8).14(C2×C10), SmallGroup(480,942)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C8.C22
G = < a,b,c,d | a15=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >
Subgroups: 168 in 120 conjugacy classes, 80 normal (48 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C10, C10, C12, C12, C2×C6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C3×Q8, C30, C30, C8.C22, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C5×Q8, C3×M4(2), C3×SD16, C3×Q16, C6×Q8, C3×C4○D4, C60, C60, C2×C30, C2×C30, C5×M4(2), C5×SD16, C5×Q16, Q8×C10, C5×C4○D4, C3×C8.C22, C120, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, Q8×C15, Q8×C15, C5×C8.C22, C15×M4(2), C15×SD16, C15×Q16, Q8×C30, C15×C4○D4, C15×C8.C22
Quotients: C1, C2, C3, C22, C5, C6, D4, C23, C10, C2×C6, C15, C2×D4, C2×C10, C3×D4, C22×C6, C30, C8.C22, C5×D4, C22×C10, C6×D4, C2×C30, D4×C10, C3×C8.C22, D4×C15, C22×C30, C5×C8.C22, D4×C30, C15×C8.C22
►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 68 40 158 232 168 85 17)(2 69 41 159 233 169 86 18)(3 70 42 160 234 170 87 19)(4 71 43 161 235 171 88 20)(5 72 44 162 236 172 89 21)(6 73 45 163 237 173 90 22)(7 74 31 164 238 174 76 23)(8 75 32 165 239 175 77 24)(9 61 33 151 240 176 78 25)(10 62 34 152 226 177 79 26)(11 63 35 153 227 178 80 27)(12 64 36 154 228 179 81 28)(13 65 37 155 229 180 82 29)(14 66 38 156 230 166 83 30)(15 67 39 157 231 167 84 16)(46 211 198 189 97 106 143 132)(47 212 199 190 98 107 144 133)(48 213 200 191 99 108 145 134)(49 214 201 192 100 109 146 135)(50 215 202 193 101 110 147 121)(51 216 203 194 102 111 148 122)(52 217 204 195 103 112 149 123)(53 218 205 181 104 113 150 124)(54 219 206 182 105 114 136 125)(55 220 207 183 91 115 137 126)(56 221 208 184 92 116 138 127)(57 222 209 185 93 117 139 128)(58 223 210 186 94 118 140 129)(59 224 196 187 95 119 141 130)(60 225 197 188 96 120 142 131)
(1 186)(2 187)(3 188)(4 189)(5 190)(6 191)(7 192)(8 193)(9 194)(10 195)(11 181)(12 182)(13 183)(14 184)(15 185)(16 57)(17 58)(18 59)(19 60)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 214)(32 215)(33 216)(34 217)(35 218)(36 219)(37 220)(38 221)(39 222)(40 223)(41 224)(42 225)(43 211)(44 212)(45 213)(61 148)(62 149)(63 150)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)(73 145)(74 146)(75 147)(76 109)(77 110)(78 111)(79 112)(80 113)(81 114)(82 115)(83 116)(84 117)(85 118)(86 119)(87 120)(88 106)(89 107)(90 108)(91 155)(92 156)(93 157)(94 158)(95 159)(96 160)(97 161)(98 162)(99 163)(100 164)(101 165)(102 151)(103 152)(104 153)(105 154)(121 239)(122 240)(123 226)(124 227)(125 228)(126 229)(127 230)(128 231)(129 232)(130 233)(131 234)(132 235)(133 236)(134 237)(135 238)(166 208)(167 209)(168 210)(169 196)(170 197)(171 198)(172 199)(173 200)(174 201)(175 202)(176 203)(177 204)(178 205)(179 206)(180 207)
(16 157)(17 158)(18 159)(19 160)(20 161)(21 162)(22 163)(23 164)(24 165)(25 151)(26 152)(27 153)(28 154)(29 155)(30 156)(61 176)(62 177)(63 178)(64 179)(65 180)(66 166)(67 167)(68 168)(69 169)(70 170)(71 171)(72 172)(73 173)(74 174)(75 175)(106 211)(107 212)(108 213)(109 214)(110 215)(111 216)(112 217)(113 218)(114 219)(115 220)(116 221)(117 222)(118 223)(119 224)(120 225)(121 193)(122 194)(123 195)(124 181)(125 182)(126 183)(127 184)(128 185)(129 186)(130 187)(131 188)(132 189)(133 190)(134 191)(135 192)
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), (1,68,40,158,232,168,85,17)(2,69,41,159,233,169,86,18)(3,70,42,160,234,170,87,19)(4,71,43,161,235,171,88,20)(5,72,44,162,236,172,89,21)(6,73,45,163,237,173,90,22)(7,74,31,164,238,174,76,23)(8,75,32,165,239,175,77,24)(9,61,33,151,240,176,78,25)(10,62,34,152,226,177,79,26)(11,63,35,153,227,178,80,27)(12,64,36,154,228,179,81,28)(13,65,37,155,229,180,82,29)(14,66,38,156,230,166,83,30)(15,67,39,157,231,167,84,16)(46,211,198,189,97,106,143,132)(47,212,199,190,98,107,144,133)(48,213,200,191,99,108,145,134)(49,214,201,192,100,109,146,135)(50,215,202,193,101,110,147,121)(51,216,203,194,102,111,148,122)(52,217,204,195,103,112,149,123)(53,218,205,181,104,113,150,124)(54,219,206,182,105,114,136,125)(55,220,207,183,91,115,137,126)(56,221,208,184,92,116,138,127)(57,222,209,185,93,117,139,128)(58,223,210,186,94,118,140,129)(59,224,196,187,95,119,141,130)(60,225,197,188,96,120,142,131), (1,186)(2,187)(3,188)(4,189)(5,190)(6,191)(7,192)(8,193)(9,194)(10,195)(11,181)(12,182)(13,183)(14,184)(15,185)(16,57)(17,58)(18,59)(19,60)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,214)(32,215)(33,216)(34,217)(35,218)(36,219)(37,220)(38,221)(39,222)(40,223)(41,224)(42,225)(43,211)(44,212)(45,213)(61,148)(62,149)(63,150)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144)(73,145)(74,146)(75,147)(76,109)(77,110)(78,111)(79,112)(80,113)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,106)(89,107)(90,108)(91,155)(92,156)(93,157)(94,158)(95,159)(96,160)(97,161)(98,162)(99,163)(100,164)(101,165)(102,151)(103,152)(104,153)(105,154)(121,239)(122,240)(123,226)(124,227)(125,228)(126,229)(127,230)(128,231)(129,232)(130,233)(131,234)(132,235)(133,236)(134,237)(135,238)(166,208)(167,209)(168,210)(169,196)(170,197)(171,198)(172,199)(173,200)(174,201)(175,202)(176,203)(177,204)(178,205)(179,206)(180,207), (16,157)(17,158)(18,159)(19,160)(20,161)(21,162)(22,163)(23,164)(24,165)(25,151)(26,152)(27,153)(28,154)(29,155)(30,156)(61,176)(62,177)(63,178)(64,179)(65,180)(66,166)(67,167)(68,168)(69,169)(70,170)(71,171)(72,172)(73,173)(74,174)(75,175)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217)(113,218)(114,219)(115,220)(116,221)(117,222)(118,223)(119,224)(120,225)(121,193)(122,194)(123,195)(124,181)(125,182)(126,183)(127,184)(128,185)(129,186)(130,187)(131,188)(132,189)(133,190)(134,191)(135,192)>;
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), (1,68,40,158,232,168,85,17)(2,69,41,159,233,169,86,18)(3,70,42,160,234,170,87,19)(4,71,43,161,235,171,88,20)(5,72,44,162,236,172,89,21)(6,73,45,163,237,173,90,22)(7,74,31,164,238,174,76,23)(8,75,32,165,239,175,77,24)(9,61,33,151,240,176,78,25)(10,62,34,152,226,177,79,26)(11,63,35,153,227,178,80,27)(12,64,36,154,228,179,81,28)(13,65,37,155,229,180,82,29)(14,66,38,156,230,166,83,30)(15,67,39,157,231,167,84,16)(46,211,198,189,97,106,143,132)(47,212,199,190,98,107,144,133)(48,213,200,191,99,108,145,134)(49,214,201,192,100,109,146,135)(50,215,202,193,101,110,147,121)(51,216,203,194,102,111,148,122)(52,217,204,195,103,112,149,123)(53,218,205,181,104,113,150,124)(54,219,206,182,105,114,136,125)(55,220,207,183,91,115,137,126)(56,221,208,184,92,116,138,127)(57,222,209,185,93,117,139,128)(58,223,210,186,94,118,140,129)(59,224,196,187,95,119,141,130)(60,225,197,188,96,120,142,131), (1,186)(2,187)(3,188)(4,189)(5,190)(6,191)(7,192)(8,193)(9,194)(10,195)(11,181)(12,182)(13,183)(14,184)(15,185)(16,57)(17,58)(18,59)(19,60)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,214)(32,215)(33,216)(34,217)(35,218)(36,219)(37,220)(38,221)(39,222)(40,223)(41,224)(42,225)(43,211)(44,212)(45,213)(61,148)(62,149)(63,150)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144)(73,145)(74,146)(75,147)(76,109)(77,110)(78,111)(79,112)(80,113)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,106)(89,107)(90,108)(91,155)(92,156)(93,157)(94,158)(95,159)(96,160)(97,161)(98,162)(99,163)(100,164)(101,165)(102,151)(103,152)(104,153)(105,154)(121,239)(122,240)(123,226)(124,227)(125,228)(126,229)(127,230)(128,231)(129,232)(130,233)(131,234)(132,235)(133,236)(134,237)(135,238)(166,208)(167,209)(168,210)(169,196)(170,197)(171,198)(172,199)(173,200)(174,201)(175,202)(176,203)(177,204)(178,205)(179,206)(180,207), (16,157)(17,158)(18,159)(19,160)(20,161)(21,162)(22,163)(23,164)(24,165)(25,151)(26,152)(27,153)(28,154)(29,155)(30,156)(61,176)(62,177)(63,178)(64,179)(65,180)(66,166)(67,167)(68,168)(69,169)(70,170)(71,171)(72,172)(73,173)(74,174)(75,175)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217)(113,218)(114,219)(115,220)(116,221)(117,222)(118,223)(119,224)(120,225)(121,193)(122,194)(123,195)(124,181)(125,182)(126,183)(127,184)(128,185)(129,186)(130,187)(131,188)(132,189)(133,190)(134,191)(135,192) );
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)], [(1,68,40,158,232,168,85,17),(2,69,41,159,233,169,86,18),(3,70,42,160,234,170,87,19),(4,71,43,161,235,171,88,20),(5,72,44,162,236,172,89,21),(6,73,45,163,237,173,90,22),(7,74,31,164,238,174,76,23),(8,75,32,165,239,175,77,24),(9,61,33,151,240,176,78,25),(10,62,34,152,226,177,79,26),(11,63,35,153,227,178,80,27),(12,64,36,154,228,179,81,28),(13,65,37,155,229,180,82,29),(14,66,38,156,230,166,83,30),(15,67,39,157,231,167,84,16),(46,211,198,189,97,106,143,132),(47,212,199,190,98,107,144,133),(48,213,200,191,99,108,145,134),(49,214,201,192,100,109,146,135),(50,215,202,193,101,110,147,121),(51,216,203,194,102,111,148,122),(52,217,204,195,103,112,149,123),(53,218,205,181,104,113,150,124),(54,219,206,182,105,114,136,125),(55,220,207,183,91,115,137,126),(56,221,208,184,92,116,138,127),(57,222,209,185,93,117,139,128),(58,223,210,186,94,118,140,129),(59,224,196,187,95,119,141,130),(60,225,197,188,96,120,142,131)], [(1,186),(2,187),(3,188),(4,189),(5,190),(6,191),(7,192),(8,193),(9,194),(10,195),(11,181),(12,182),(13,183),(14,184),(15,185),(16,57),(17,58),(18,59),(19,60),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(31,214),(32,215),(33,216),(34,217),(35,218),(36,219),(37,220),(38,221),(39,222),(40,223),(41,224),(42,225),(43,211),(44,212),(45,213),(61,148),(62,149),(63,150),(64,136),(65,137),(66,138),(67,139),(68,140),(69,141),(70,142),(71,143),(72,144),(73,145),(74,146),(75,147),(76,109),(77,110),(78,111),(79,112),(80,113),(81,114),(82,115),(83,116),(84,117),(85,118),(86,119),(87,120),(88,106),(89,107),(90,108),(91,155),(92,156),(93,157),(94,158),(95,159),(96,160),(97,161),(98,162),(99,163),(100,164),(101,165),(102,151),(103,152),(104,153),(105,154),(121,239),(122,240),(123,226),(124,227),(125,228),(126,229),(127,230),(128,231),(129,232),(130,233),(131,234),(132,235),(133,236),(134,237),(135,238),(166,208),(167,209),(168,210),(169,196),(170,197),(171,198),(172,199),(173,200),(174,201),(175,202),(176,203),(177,204),(178,205),(179,206),(180,207)], [(16,157),(17,158),(18,159),(19,160),(20,161),(21,162),(22,163),(23,164),(24,165),(25,151),(26,152),(27,153),(28,154),(29,155),(30,156),(61,176),(62,177),(63,178),(64,179),(65,180),(66,166),(67,167),(68,168),(69,169),(70,170),(71,171),(72,172),(73,173),(74,174),(75,175),(106,211),(107,212),(108,213),(109,214),(110,215),(111,216),(112,217),(113,218),(114,219),(115,220),(116,221),(117,222),(118,223),(119,224),(120,225),(121,193),(122,194),(123,195),(124,181),(125,182),(126,183),(127,184),(128,185),(129,186),(130,187),(131,188),(132,189),(133,190),(134,191),(135,192)]])
165 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 15A | ··· | 15H | 20A | ··· | 20H | 20I | ··· | 20T | 24A | 24B | 24C | 24D | 30A | ··· | 30H | 30I | ··· | 30P | 30Q | ··· | 30X | 40A | ··· | 40H | 60A | ··· | 60P | 60Q | ··· | 60AN | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 30 | ··· | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
165 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | |||||||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C5 | C6 | C6 | C6 | C6 | C6 | C10 | C10 | C10 | C10 | C10 | C15 | C30 | C30 | C30 | C30 | C30 | D4 | D4 | C3×D4 | C3×D4 | C5×D4 | C5×D4 | D4×C15 | D4×C15 | C8.C22 | C3×C8.C22 | C5×C8.C22 | C15×C8.C22 |
| kernel | C15×C8.C22 | C15×M4(2) | C15×SD16 | C15×Q16 | Q8×C30 | C15×C4○D4 | C5×C8.C22 | C3×C8.C22 | C5×M4(2) | C5×SD16 | C5×Q16 | Q8×C10 | C5×C4○D4 | C3×M4(2) | C3×SD16 | C3×Q16 | C6×Q8 | C3×C4○D4 | C8.C22 | M4(2) | SD16 | Q16 | C2×Q8 | C4○D4 | C60 | C2×C30 | C20 | C2×C10 | C12 | C2×C6 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 2 | 2 | 4 | 8 | 8 | 4 | 4 | 8 | 8 | 16 | 16 | 8 | 8 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C15×C8.C22 ►in GL6(𝔽241)
| 225 | 0 | 0 | 0 | 0 | 0 |
| 0 | 225 | 0 | 0 | 0 | 0 |
| 0 | 0 | 205 | 0 | 0 | 0 |
| 0 | 0 | 0 | 205 | 0 | 0 |
| 0 | 0 | 0 | 0 | 205 | 0 |
| 0 | 0 | 0 | 0 | 0 | 205 |
| 184 | 57 | 0 | 0 | 0 | 0 |
| 36 | 57 | 0 | 0 | 0 | 0 |
| 0 | 0 | 158 | 2 | 85 | 0 |
| 0 | 0 | 0 | 0 | 239 | 85 |
| 0 | 0 | 4 | 87 | 239 | 239 |
| 0 | 0 | 71 | 158 | 85 | 85 |
| 184 | 57 | 0 | 0 | 0 | 0 |
| 91 | 57 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 239 | 240 | 1 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 240 | 240 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
G:=sub<GL(6,GF(241))| [225,0,0,0,0,0,0,225,0,0,0,0,0,0,205,0,0,0,0,0,0,205,0,0,0,0,0,0,205,0,0,0,0,0,0,205],[184,36,0,0,0,0,57,57,0,0,0,0,0,0,158,0,4,71,0,0,2,0,87,158,0,0,85,239,239,85,0,0,0,85,239,85],[184,91,0,0,0,0,57,57,0,0,0,0,0,0,240,0,239,0,0,0,0,0,240,1,0,0,0,0,1,0,0,0,0,1,1,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,240,0,240,0,0,0,240,0,0,240] >;
C15×C8.C22 in GAP, Magma, Sage, TeX
C_{15}\times C_8.C_2^2 % in TeX
G:=Group("C15xC8.C2^2"); // GroupNames label
G:=SmallGroup(480,942);
// by ID
G=gap.SmallGroup(480,942);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1709,1688,5126,15125,7572,124]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
// generators/relations