C2^2xDic15 - GroupNames

G = C₂²×Dic₁₅ order 240 = 2⁴·3·5

Direct product of C₂² and Dic₁₅

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

Derived series

C₁ — C₁₅ — C₂²×Dic₁₅

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — Dic₁₅ — C₂×Dic₁₅ — C₂²×Dic₁₅

Lower central

C₁₅ — C₂²×Dic₁₅

Upper central

C₁ — C₂³

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, C₃, C₄, C₂², C₅, C₆, C₆, C₂×C₄, C₂³, C₁₀, C₁₀, Dic₃, C₂×C₆, C₁₅, C₂²×C₄, Dic₅, C₂×C₁₀, C₂×Dic₃, C₂²×C₆, C₃₀, C₃₀, C₂×Dic₅, C₂²×C₁₀, C₂²×Dic₃, Dic₁₅, C₂×C₃₀, C₂²×Dic₅, C₂×Dic₁₅, C₂²×C₃₀, C₂²×Dic₁₅
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, C₂³, D₅, Dic₃, D₆, C₂²×C₄, Dic₅, D₁₀, C₂×Dic₃, C₂²×S₃, D₁₅, C₂×Dic₅, C₂²×D₅, C₂²×Dic₃, Dic₁₅, D₃₀, C₂²×Dic₅, C₂×Dic₁₅, C₂²×D₁₅, C₂²×Dic₁₅

Smallest permutation representation of C₂²×Dic₁₅
►Regular action on 240 points

Generators in S₂₄₀

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

(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)

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

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

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

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

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

G = C22×Dic15 order 240 = 24·3·5

Direct product of C22 and Dic15

G = C₂²×Dic₁₅ order 240 = 2⁴·3·5

Direct product of C₂² and Dic₁₅