direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×Dic15, C23.3D15, C30.36C23, C22.11D30, (C2×C30)⋊7C4, C30⋊10(C2×C4), C6⋊2(C2×Dic5), (C2×C6)⋊3Dic5, C15⋊11(C22×C4), (C2×C10)⋊8Dic3, C10⋊3(C2×Dic3), (C2×C6).30D10, (C2×C10).30D6, (C22×C6).3D5, C5⋊3(C22×Dic3), C3⋊2(C22×Dic5), (C22×C10).5S3, (C22×C30).3C2, C2.2(C22×D15), C6.36(C22×D5), (C2×C30).31C22, C10.36(C22×S3), SmallGroup(240,183)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C22×Dic15 |
Generators and relations for C22×Dic15
G = < a,b,c,d | a2=b2=c30=1, d2=c15, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 328 in 108 conjugacy classes, 75 normal (13 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C22 [×7], C5, C6, C6 [×6], C2×C4 [×6], C23, C10, C10 [×6], Dic3 [×4], C2×C6 [×7], C15, C22×C4, Dic5 [×4], C2×C10 [×7], C2×Dic3 [×6], C22×C6, C30, C30 [×6], C2×Dic5 [×6], C22×C10, C22×Dic3, Dic15 [×4], C2×C30 [×7], C22×Dic5, C2×Dic15 [×6], C22×C30, C22×Dic15
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, Dic3 [×4], D6 [×3], C22×C4, Dic5 [×4], D10 [×3], C2×Dic3 [×6], C22×S3, D15, C2×Dic5 [×6], C22×D5, C22×Dic3, Dic15 [×4], D30 [×3], C22×Dic5, C2×Dic15 [×6], C22×D15, C22×Dic15
►Regular action on 240 points
Generators in S240
(1 106)(2 107)(3 108)(4 109)(5 110)(6 111)(7 112)(8 113)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 120)(16 91)(17 92)(18 93)(19 94)(20 95)(21 96)(22 97)(23 98)(24 99)(25 100)(26 101)(27 102)(28 103)(29 104)(30 105)(31 70)(32 71)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(41 80)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 88)(50 89)(51 90)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(121 216)(122 217)(123 218)(124 219)(125 220)(126 221)(127 222)(128 223)(129 224)(130 225)(131 226)(132 227)(133 228)(134 229)(135 230)(136 231)(137 232)(138 233)(139 234)(140 235)(141 236)(142 237)(143 238)(144 239)(145 240)(146 211)(147 212)(148 213)(149 214)(150 215)(151 206)(152 207)(153 208)(154 209)(155 210)(156 181)(157 182)(158 183)(159 184)(160 185)(161 186)(162 187)(163 188)(164 189)(165 190)(166 191)(167 192)(168 193)(169 194)(170 195)(171 196)(172 197)(173 198)(174 199)(175 200)(176 201)(177 202)(178 203)(179 204)(180 205)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(61 114)(62 115)(63 116)(64 117)(65 118)(66 119)(67 120)(68 91)(69 92)(70 93)(71 94)(72 95)(73 96)(74 97)(75 98)(76 99)(77 100)(78 101)(79 102)(80 103)(81 104)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(89 112)(90 113)(121 161)(122 162)(123 163)(124 164)(125 165)(126 166)(127 167)(128 168)(129 169)(130 170)(131 171)(132 172)(133 173)(134 174)(135 175)(136 176)(137 177)(138 178)(139 179)(140 180)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)(181 211)(182 212)(183 213)(184 214)(185 215)(186 216)(187 217)(188 218)(189 219)(190 220)(191 221)(192 222)(193 223)(194 224)(195 225)(196 226)(197 227)(198 228)(199 229)(200 230)(201 231)(202 232)(203 233)(204 234)(205 235)(206 236)(207 237)(208 238)(209 239)(210 240)
(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)
(1 176 16 161)(2 175 17 160)(3 174 18 159)(4 173 19 158)(5 172 20 157)(6 171 21 156)(7 170 22 155)(8 169 23 154)(9 168 24 153)(10 167 25 152)(11 166 26 151)(12 165 27 180)(13 164 28 179)(14 163 29 178)(15 162 30 177)(31 149 46 134)(32 148 47 133)(33 147 48 132)(34 146 49 131)(35 145 50 130)(36 144 51 129)(37 143 52 128)(38 142 53 127)(39 141 54 126)(40 140 55 125)(41 139 56 124)(42 138 57 123)(43 137 58 122)(44 136 59 121)(45 135 60 150)(61 223 76 238)(62 222 77 237)(63 221 78 236)(64 220 79 235)(65 219 80 234)(66 218 81 233)(67 217 82 232)(68 216 83 231)(69 215 84 230)(70 214 85 229)(71 213 86 228)(72 212 87 227)(73 211 88 226)(74 240 89 225)(75 239 90 224)(91 186 106 201)(92 185 107 200)(93 184 108 199)(94 183 109 198)(95 182 110 197)(96 181 111 196)(97 210 112 195)(98 209 113 194)(99 208 114 193)(100 207 115 192)(101 206 116 191)(102 205 117 190)(103 204 118 189)(104 203 119 188)(105 202 120 187)
G:=sub<Sym(240)| (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,113)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,120)(16,91)(17,92)(18,93)(19,94)(20,95)(21,96)(22,97)(23,98)(24,99)(25,100)(26,101)(27,102)(28,103)(29,104)(30,105)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(121,216)(122,217)(123,218)(124,219)(125,220)(126,221)(127,222)(128,223)(129,224)(130,225)(131,226)(132,227)(133,228)(134,229)(135,230)(136,231)(137,232)(138,233)(139,234)(140,235)(141,236)(142,237)(143,238)(144,239)(145,240)(146,211)(147,212)(148,213)(149,214)(150,215)(151,206)(152,207)(153,208)(154,209)(155,210)(156,181)(157,182)(158,183)(159,184)(160,185)(161,186)(162,187)(163,188)(164,189)(165,190)(166,191)(167,192)(168,193)(169,194)(170,195)(171,196)(172,197)(173,198)(174,199)(175,200)(176,201)(177,202)(178,203)(179,204)(180,205), (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,91)(69,92)(70,93)(71,94)(72,95)(73,96)(74,97)(75,98)(76,99)(77,100)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(121,161)(122,162)(123,163)(124,164)(125,165)(126,166)(127,167)(128,168)(129,169)(130,170)(131,171)(132,172)(133,173)(134,174)(135,175)(136,176)(137,177)(138,178)(139,179)(140,180)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160)(181,211)(182,212)(183,213)(184,214)(185,215)(186,216)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(199,229)(200,230)(201,231)(202,232)(203,233)(204,234)(205,235)(206,236)(207,237)(208,238)(209,239)(210,240), (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), (1,176,16,161)(2,175,17,160)(3,174,18,159)(4,173,19,158)(5,172,20,157)(6,171,21,156)(7,170,22,155)(8,169,23,154)(9,168,24,153)(10,167,25,152)(11,166,26,151)(12,165,27,180)(13,164,28,179)(14,163,29,178)(15,162,30,177)(31,149,46,134)(32,148,47,133)(33,147,48,132)(34,146,49,131)(35,145,50,130)(36,144,51,129)(37,143,52,128)(38,142,53,127)(39,141,54,126)(40,140,55,125)(41,139,56,124)(42,138,57,123)(43,137,58,122)(44,136,59,121)(45,135,60,150)(61,223,76,238)(62,222,77,237)(63,221,78,236)(64,220,79,235)(65,219,80,234)(66,218,81,233)(67,217,82,232)(68,216,83,231)(69,215,84,230)(70,214,85,229)(71,213,86,228)(72,212,87,227)(73,211,88,226)(74,240,89,225)(75,239,90,224)(91,186,106,201)(92,185,107,200)(93,184,108,199)(94,183,109,198)(95,182,110,197)(96,181,111,196)(97,210,112,195)(98,209,113,194)(99,208,114,193)(100,207,115,192)(101,206,116,191)(102,205,117,190)(103,204,118,189)(104,203,119,188)(105,202,120,187)>;
G:=Group( (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,113)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,120)(16,91)(17,92)(18,93)(19,94)(20,95)(21,96)(22,97)(23,98)(24,99)(25,100)(26,101)(27,102)(28,103)(29,104)(30,105)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(121,216)(122,217)(123,218)(124,219)(125,220)(126,221)(127,222)(128,223)(129,224)(130,225)(131,226)(132,227)(133,228)(134,229)(135,230)(136,231)(137,232)(138,233)(139,234)(140,235)(141,236)(142,237)(143,238)(144,239)(145,240)(146,211)(147,212)(148,213)(149,214)(150,215)(151,206)(152,207)(153,208)(154,209)(155,210)(156,181)(157,182)(158,183)(159,184)(160,185)(161,186)(162,187)(163,188)(164,189)(165,190)(166,191)(167,192)(168,193)(169,194)(170,195)(171,196)(172,197)(173,198)(174,199)(175,200)(176,201)(177,202)(178,203)(179,204)(180,205), (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,91)(69,92)(70,93)(71,94)(72,95)(73,96)(74,97)(75,98)(76,99)(77,100)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(121,161)(122,162)(123,163)(124,164)(125,165)(126,166)(127,167)(128,168)(129,169)(130,170)(131,171)(132,172)(133,173)(134,174)(135,175)(136,176)(137,177)(138,178)(139,179)(140,180)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160)(181,211)(182,212)(183,213)(184,214)(185,215)(186,216)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222)(193,223)(194,224)(195,225)(196,226)(197,227)(198,228)(199,229)(200,230)(201,231)(202,232)(203,233)(204,234)(205,235)(206,236)(207,237)(208,238)(209,239)(210,240), (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), (1,176,16,161)(2,175,17,160)(3,174,18,159)(4,173,19,158)(5,172,20,157)(6,171,21,156)(7,170,22,155)(8,169,23,154)(9,168,24,153)(10,167,25,152)(11,166,26,151)(12,165,27,180)(13,164,28,179)(14,163,29,178)(15,162,30,177)(31,149,46,134)(32,148,47,133)(33,147,48,132)(34,146,49,131)(35,145,50,130)(36,144,51,129)(37,143,52,128)(38,142,53,127)(39,141,54,126)(40,140,55,125)(41,139,56,124)(42,138,57,123)(43,137,58,122)(44,136,59,121)(45,135,60,150)(61,223,76,238)(62,222,77,237)(63,221,78,236)(64,220,79,235)(65,219,80,234)(66,218,81,233)(67,217,82,232)(68,216,83,231)(69,215,84,230)(70,214,85,229)(71,213,86,228)(72,212,87,227)(73,211,88,226)(74,240,89,225)(75,239,90,224)(91,186,106,201)(92,185,107,200)(93,184,108,199)(94,183,109,198)(95,182,110,197)(96,181,111,196)(97,210,112,195)(98,209,113,194)(99,208,114,193)(100,207,115,192)(101,206,116,191)(102,205,117,190)(103,204,118,189)(104,203,119,188)(105,202,120,187) );
G=PermutationGroup([(1,106),(2,107),(3,108),(4,109),(5,110),(6,111),(7,112),(8,113),(9,114),(10,115),(11,116),(12,117),(13,118),(14,119),(15,120),(16,91),(17,92),(18,93),(19,94),(20,95),(21,96),(22,97),(23,98),(24,99),(25,100),(26,101),(27,102),(28,103),(29,104),(30,105),(31,70),(32,71),(33,72),(34,73),(35,74),(36,75),(37,76),(38,77),(39,78),(40,79),(41,80),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,88),(50,89),(51,90),(52,61),(53,62),(54,63),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(121,216),(122,217),(123,218),(124,219),(125,220),(126,221),(127,222),(128,223),(129,224),(130,225),(131,226),(132,227),(133,228),(134,229),(135,230),(136,231),(137,232),(138,233),(139,234),(140,235),(141,236),(142,237),(143,238),(144,239),(145,240),(146,211),(147,212),(148,213),(149,214),(150,215),(151,206),(152,207),(153,208),(154,209),(155,210),(156,181),(157,182),(158,183),(159,184),(160,185),(161,186),(162,187),(163,188),(164,189),(165,190),(166,191),(167,192),(168,193),(169,194),(170,195),(171,196),(172,197),(173,198),(174,199),(175,200),(176,201),(177,202),(178,203),(179,204),(180,205)], [(1,44),(2,45),(3,46),(4,47),(5,48),(6,49),(7,50),(8,51),(9,52),(10,53),(11,54),(12,55),(13,56),(14,57),(15,58),(16,59),(17,60),(18,31),(19,32),(20,33),(21,34),(22,35),(23,36),(24,37),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(61,114),(62,115),(63,116),(64,117),(65,118),(66,119),(67,120),(68,91),(69,92),(70,93),(71,94),(72,95),(73,96),(74,97),(75,98),(76,99),(77,100),(78,101),(79,102),(80,103),(81,104),(82,105),(83,106),(84,107),(85,108),(86,109),(87,110),(88,111),(89,112),(90,113),(121,161),(122,162),(123,163),(124,164),(125,165),(126,166),(127,167),(128,168),(129,169),(130,170),(131,171),(132,172),(133,173),(134,174),(135,175),(136,176),(137,177),(138,178),(139,179),(140,180),(141,151),(142,152),(143,153),(144,154),(145,155),(146,156),(147,157),(148,158),(149,159),(150,160),(181,211),(182,212),(183,213),(184,214),(185,215),(186,216),(187,217),(188,218),(189,219),(190,220),(191,221),(192,222),(193,223),(194,224),(195,225),(196,226),(197,227),(198,228),(199,229),(200,230),(201,231),(202,232),(203,233),(204,234),(205,235),(206,236),(207,237),(208,238),(209,239),(210,240)], [(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)], [(1,176,16,161),(2,175,17,160),(3,174,18,159),(4,173,19,158),(5,172,20,157),(6,171,21,156),(7,170,22,155),(8,169,23,154),(9,168,24,153),(10,167,25,152),(11,166,26,151),(12,165,27,180),(13,164,28,179),(14,163,29,178),(15,162,30,177),(31,149,46,134),(32,148,47,133),(33,147,48,132),(34,146,49,131),(35,145,50,130),(36,144,51,129),(37,143,52,128),(38,142,53,127),(39,141,54,126),(40,140,55,125),(41,139,56,124),(42,138,57,123),(43,137,58,122),(44,136,59,121),(45,135,60,150),(61,223,76,238),(62,222,77,237),(63,221,78,236),(64,220,79,235),(65,219,80,234),(66,218,81,233),(67,217,82,232),(68,216,83,231),(69,215,84,230),(70,214,85,229),(71,213,86,228),(72,212,87,227),(73,211,88,226),(74,240,89,225),(75,239,90,224),(91,186,106,201),(92,185,107,200),(93,184,108,199),(94,183,109,198),(95,182,110,197),(96,181,111,196),(97,210,112,195),(98,209,113,194),(99,208,114,193),(100,207,115,192),(101,206,116,191),(102,205,117,190),(103,204,118,189),(104,203,119,188),(105,202,120,187)])
C22×Dic15 is a maximal subgroup of
C30.24C42 C30.29C42 C2×Dic3×Dic5 C23.48(S3×D5) C6.(C2×D20) (C2×C10).D12 Dic15⋊16D4 Dic15⋊17D4 Dic15⋊18D4 Dic15.48D4 C23.15D30 C22⋊2Dic30 Dic15⋊19D4 C22.D60 C23.22D30 Dic15⋊12D4 C22×D5×Dic3 C22×S3×Dic5 C22×C4×D15
C22×Dic15 is a maximal quotient of
C23.26D30 D4.Dic15
72 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | ··· | 4H | 5A | 5B | 6A | ··· | 6G | 10A | ··· | 10N | 15A | 15B | 15C | 15D | 30A | ··· | 30AB |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | ··· | 1 | 2 | 15 | ··· | 15 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | S3 | D5 | Dic3 | D6 | Dic5 | D10 | D15 | Dic15 | D30 |
| kernel | C22×Dic15 | C2×Dic15 | C22×C30 | C2×C30 | C22×C10 | C22×C6 | C2×C10 | C2×C10 | C2×C6 | C2×C6 | C23 | C22 | C22 |
| # reps | 1 | 6 | 1 | 8 | 1 | 2 | 4 | 3 | 8 | 6 | 4 | 16 | 12 |
Matrix representation of C22×Dic15 ►in GL7(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 45 | 8 |
| 0 | 0 | 0 | 0 | 0 | 53 | 23 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 34 | 9 | 0 | 0 | 0 | 0 |
| 0 | 7 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 33 | 0 | 0 |
| 0 | 0 | 0 | 47 | 55 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 | 1 |
| 0 | 0 | 0 | 0 | 0 | 43 | 43 |
G:=sub<GL(7,GF(61))| [60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,60,18,0,0,0,0,0,0,0,0,60,0,0,0,0,0,1,43,0,0,0,0,0,0,0,45,53,0,0,0,0,0,8,23],[60,0,0,0,0,0,0,0,34,7,0,0,0,0,0,9,27,0,0,0,0,0,0,0,6,47,0,0,0,0,0,33,55,0,0,0,0,0,0,0,18,43,0,0,0,0,0,1,43] >;
C22×Dic15 in GAP, Magma, Sage, TeX
C_2^2\times {\rm Dic}_{15} % in TeX
G:=Group("C2^2xDic15"); // GroupNames label
G:=SmallGroup(240,183);
// by ID
G=gap.SmallGroup(240,183);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,964,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^30=1,d^2=c^15,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations