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## G = C22⋊2Dic30order 480 = 25·3·5

### The semidirect product of C22 and Dic30 acting via Dic30/Dic15=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C22⋊2Dic30
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C2×Dic15 — C22×Dic15 — C22⋊2Dic30
 Lower central C15 — C2×C30 — C22⋊2Dic30
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C222Dic30
G = < a,b,c,d | a2=b2=c60=1, d2=c30, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 740 in 148 conjugacy classes, 57 normal (47 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×7], C22, C22 [×2], C22 [×2], C5, C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C10 [×3], C10 [×2], Dic3 [×5], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4, C22⋊C4, C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×5], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], Dic6 [×2], C2×Dic3 [×6], C2×C12 [×2], C22×C6, C30 [×3], C30 [×2], C22⋊Q8, Dic10 [×2], C2×Dic5 [×6], C2×C20 [×2], C22×C10, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4, C3×C22⋊C4, C2×Dic6, C22×Dic3, Dic15 [×2], Dic15 [×3], C60 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C10.D4 [×2], C4⋊Dic5, C23.D5, C5×C22⋊C4, C2×Dic10, C22×Dic5, Dic3.D4, Dic30 [×2], C2×Dic15 [×4], C2×Dic15 [×2], C2×C60 [×2], C22×C30, Dic5.14D4, C30.4Q8 [×2], C605C4, C30.38D4, C15×C22⋊C4, C2×Dic30, C22×Dic15, C222Dic30
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D5, D6 [×3], C2×D4, C2×Q8, C4○D4, D10 [×3], Dic6 [×2], C22×S3, D15, C22⋊Q8, Dic10 [×2], C22×D5, C2×Dic6, S3×D4, D42S3, D30 [×3], C2×Dic10, D4×D5, D42D5, Dic3.D4, Dic30 [×2], C22×D15, Dic5.14D4, C2×Dic30, D4×D15, D42D15, C222Dic30

Smallest permutation representation of C222Dic30
On 240 points
Generators in S240
(1 31)(2 76)(3 33)(4 78)(5 35)(6 80)(7 37)(8 82)(9 39)(10 84)(11 41)(12 86)(13 43)(14 88)(15 45)(16 90)(17 47)(18 92)(19 49)(20 94)(21 51)(22 96)(23 53)(24 98)(25 55)(26 100)(27 57)(28 102)(29 59)(30 104)(32 106)(34 108)(36 110)(38 112)(40 114)(42 116)(44 118)(46 120)(48 62)(50 64)(52 66)(54 68)(56 70)(58 72)(60 74)(61 91)(63 93)(65 95)(67 97)(69 99)(71 101)(73 103)(75 105)(77 107)(79 109)(81 111)(83 113)(85 115)(87 117)(89 119)(121 238)(122 152)(123 240)(124 154)(125 182)(126 156)(127 184)(128 158)(129 186)(130 160)(131 188)(132 162)(133 190)(134 164)(135 192)(136 166)(137 194)(138 168)(139 196)(140 170)(141 198)(142 172)(143 200)(144 174)(145 202)(146 176)(147 204)(148 178)(149 206)(150 180)(151 208)(153 210)(155 212)(157 214)(159 216)(161 218)(163 220)(165 222)(167 224)(169 226)(171 228)(173 230)(175 232)(177 234)(179 236)(181 211)(183 213)(185 215)(187 217)(189 219)(191 221)(193 223)(195 225)(197 227)(199 229)(201 231)(203 233)(205 235)(207 237)(209 239)
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 81)(38 82)(39 83)(40 84)(41 85)(42 86)(43 87)(44 88)(45 89)(46 90)(47 91)(48 92)(49 93)(50 94)(51 95)(52 96)(53 97)(54 98)(55 99)(56 100)(57 101)(58 102)(59 103)(60 104)(121 208)(122 209)(123 210)(124 211)(125 212)(126 213)(127 214)(128 215)(129 216)(130 217)(131 218)(132 219)(133 220)(134 221)(135 222)(136 223)(137 224)(138 225)(139 226)(140 227)(141 228)(142 229)(143 230)(144 231)(145 232)(146 233)(147 234)(148 235)(149 236)(150 237)(151 238)(152 239)(153 240)(154 181)(155 182)(156 183)(157 184)(158 185)(159 186)(160 187)(161 188)(162 189)(163 190)(164 191)(165 192)(166 193)(167 194)(168 195)(169 196)(170 197)(171 198)(172 199)(173 200)(174 201)(175 202)(176 203)(177 204)(178 205)(179 206)(180 207)
(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)
(1 144 31 174)(2 143 32 173)(3 142 33 172)(4 141 34 171)(5 140 35 170)(6 139 36 169)(7 138 37 168)(8 137 38 167)(9 136 39 166)(10 135 40 165)(11 134 41 164)(12 133 42 163)(13 132 43 162)(14 131 44 161)(15 130 45 160)(16 129 46 159)(17 128 47 158)(18 127 48 157)(19 126 49 156)(20 125 50 155)(21 124 51 154)(22 123 52 153)(23 122 53 152)(24 121 54 151)(25 180 55 150)(26 179 56 149)(27 178 57 148)(28 177 58 147)(29 176 59 146)(30 175 60 145)(61 215 91 185)(62 214 92 184)(63 213 93 183)(64 212 94 182)(65 211 95 181)(66 210 96 240)(67 209 97 239)(68 208 98 238)(69 207 99 237)(70 206 100 236)(71 205 101 235)(72 204 102 234)(73 203 103 233)(74 202 104 232)(75 201 105 231)(76 200 106 230)(77 199 107 229)(78 198 108 228)(79 197 109 227)(80 196 110 226)(81 195 111 225)(82 194 112 224)(83 193 113 223)(84 192 114 222)(85 191 115 221)(86 190 116 220)(87 189 117 219)(88 188 118 218)(89 187 119 217)(90 186 120 216)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 6C 6D 6E 10A ··· 10F 10G 10H 10I 10J 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 30A ··· 30L 30M ··· 30T 60A ··· 60P order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 10 ··· 10 10 10 10 10 12 12 12 12 15 15 15 15 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 4 4 30 30 30 30 60 60 2 2 2 2 2 4 4 2 ··· 2 4 4 4 4 4 4 4 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + - + + + + + - + - + + - + - + - + - image C1 C2 C2 C2 C2 C2 C2 S3 D4 Q8 D5 D6 D6 C4○D4 D10 D10 Dic6 D15 Dic10 D30 D30 Dic30 S3×D4 D4⋊2S3 D4×D5 D4⋊2D5 D4×D15 D4⋊2D15 kernel C22⋊2Dic30 C30.4Q8 C60⋊5C4 C30.38D4 C15×C22⋊C4 C2×Dic30 C22×Dic15 C5×C22⋊C4 Dic15 C2×C30 C3×C22⋊C4 C2×C20 C22×C10 C30 C2×C12 C22×C6 C2×C10 C22⋊C4 C2×C6 C2×C4 C23 C22 C10 C10 C6 C6 C2 C2 # reps 1 2 1 1 1 1 1 1 2 2 2 2 1 2 4 2 4 4 8 8 4 16 1 1 2 2 4 4

Matrix representation of C222Dic30 in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 60 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 33 41 0 0 14 47 0 0 0 0 0 1 0 0 60 0
,
 0 57 0 0 46 0 0 0 0 0 11 0 0 0 0 50
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[33,14,0,0,41,47,0,0,0,0,0,60,0,0,1,0],[0,46,0,0,57,0,0,0,0,0,11,0,0,0,0,50] >;

C222Dic30 in GAP, Magma, Sage, TeX

C_2^2\rtimes_2{\rm Dic}_{30}
% in TeX

G:=Group("C2^2:2Dic30");
// GroupNames label

G:=SmallGroup(480,843);
// by ID

G=gap.SmallGroup(480,843);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,254,219,58,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^60=1,d^2=c^30,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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