Copied to
clipboard

## G = C3×C20⋊7D4order 480 = 25·3·5

### Direct product of C3 and C20⋊7D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×C20⋊7D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — D5×C2×C6 — C6×D20 — C3×C20⋊7D4
 Lower central C5 — C2×C10 — C3×C20⋊7D4
 Upper central C1 — C2×C6 — C22×C12

Generators and relations for C3×C207D4
G = < a,b,c,d | a3=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 672 in 188 conjugacy classes, 74 normal (42 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], C5, C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], D5 [×2], C10 [×3], C10 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×2], C2×C6 [×8], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×2], C20 [×2], C20, D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C12 [×2], C2×C12 [×4], C3×D4 [×6], C22×C6, C22×C6 [×2], C3×D5 [×2], C30 [×3], C30 [×2], C4⋊D4, D20 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×2], C22×D5 [×2], C22×C10, C3×C22⋊C4 [×2], C3×C4⋊C4, C22×C12, C6×D4 [×3], C3×Dic5 [×2], C60 [×2], C60, C6×D5 [×6], C2×C30, C2×C30 [×2], C2×C30 [×2], C4⋊Dic5, D10⋊C4 [×2], C2×D20, C2×C5⋊D4 [×2], C22×C20, C3×C4⋊D4, C3×D20 [×2], C6×Dic5 [×2], C3×C5⋊D4 [×4], C2×C60 [×2], C2×C60 [×2], D5×C2×C6 [×2], C22×C30, C207D4, C3×C4⋊Dic5, C3×D10⋊C4 [×2], C6×D20, C6×C5⋊D4 [×2], C22×C60, C3×C207D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×4], C23, D5, C2×C6 [×7], C2×D4 [×2], C4○D4, D10 [×3], C3×D4 [×4], C22×C6, C3×D5, C4⋊D4, D20 [×2], C5⋊D4 [×2], C22×D5, C6×D4 [×2], C3×C4○D4, C6×D5 [×3], C2×D20, C4○D20, C2×C5⋊D4, C3×C4⋊D4, C3×D20 [×2], C3×C5⋊D4 [×2], D5×C2×C6, C207D4, C6×D20, C3×C4○D20, C6×C5⋊D4, C3×C207D4

Smallest permutation representation of C3×C207D4
On 240 points
Generators in S240
(1 215 44)(2 216 45)(3 217 46)(4 218 47)(5 219 48)(6 220 49)(7 201 50)(8 202 51)(9 203 52)(10 204 53)(11 205 54)(12 206 55)(13 207 56)(14 208 57)(15 209 58)(16 210 59)(17 211 60)(18 212 41)(19 213 42)(20 214 43)(21 234 111)(22 235 112)(23 236 113)(24 237 114)(25 238 115)(26 239 116)(27 240 117)(28 221 118)(29 222 119)(30 223 120)(31 224 101)(32 225 102)(33 226 103)(34 227 104)(35 228 105)(36 229 106)(37 230 107)(38 231 108)(39 232 109)(40 233 110)(61 168 100)(62 169 81)(63 170 82)(64 171 83)(65 172 84)(66 173 85)(67 174 86)(68 175 87)(69 176 88)(70 177 89)(71 178 90)(72 179 91)(73 180 92)(74 161 93)(75 162 94)(76 163 95)(77 164 96)(78 165 97)(79 166 98)(80 167 99)(121 141 196)(122 142 197)(123 143 198)(124 144 199)(125 145 200)(126 146 181)(127 147 182)(128 148 183)(129 149 184)(130 150 185)(131 151 186)(132 152 187)(133 153 188)(134 154 189)(135 155 190)(136 156 191)(137 157 192)(138 158 193)(139 159 194)(140 160 195)
(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)
(1 159 25 74)(2 158 26 73)(3 157 27 72)(4 156 28 71)(5 155 29 70)(6 154 30 69)(7 153 31 68)(8 152 32 67)(9 151 33 66)(10 150 34 65)(11 149 35 64)(12 148 36 63)(13 147 37 62)(14 146 38 61)(15 145 39 80)(16 144 40 79)(17 143 21 78)(18 142 22 77)(19 141 23 76)(20 160 24 75)(41 122 112 96)(42 121 113 95)(43 140 114 94)(44 139 115 93)(45 138 116 92)(46 137 117 91)(47 136 118 90)(48 135 119 89)(49 134 120 88)(50 133 101 87)(51 132 102 86)(52 131 103 85)(53 130 104 84)(54 129 105 83)(55 128 106 82)(56 127 107 81)(57 126 108 100)(58 125 109 99)(59 124 110 98)(60 123 111 97)(161 215 194 238)(162 214 195 237)(163 213 196 236)(164 212 197 235)(165 211 198 234)(166 210 199 233)(167 209 200 232)(168 208 181 231)(169 207 182 230)(170 206 183 229)(171 205 184 228)(172 204 185 227)(173 203 186 226)(174 202 187 225)(175 201 188 224)(176 220 189 223)(177 219 190 222)(178 218 191 221)(179 217 192 240)(180 216 193 239)
(1 74)(2 73)(3 72)(4 71)(5 70)(6 69)(7 68)(8 67)(9 66)(10 65)(11 64)(12 63)(13 62)(14 61)(15 80)(16 79)(17 78)(18 77)(19 76)(20 75)(21 143)(22 142)(23 141)(24 160)(25 159)(26 158)(27 157)(28 156)(29 155)(30 154)(31 153)(32 152)(33 151)(34 150)(35 149)(36 148)(37 147)(38 146)(39 145)(40 144)(41 96)(42 95)(43 94)(44 93)(45 92)(46 91)(47 90)(48 89)(49 88)(50 87)(51 86)(52 85)(53 84)(54 83)(55 82)(56 81)(57 100)(58 99)(59 98)(60 97)(101 133)(102 132)(103 131)(104 130)(105 129)(106 128)(107 127)(108 126)(109 125)(110 124)(111 123)(112 122)(113 121)(114 140)(115 139)(116 138)(117 137)(118 136)(119 135)(120 134)(161 215)(162 214)(163 213)(164 212)(165 211)(166 210)(167 209)(168 208)(169 207)(170 206)(171 205)(172 204)(173 203)(174 202)(175 201)(176 220)(177 219)(178 218)(179 217)(180 216)(181 231)(182 230)(183 229)(184 228)(185 227)(186 226)(187 225)(188 224)(189 223)(190 222)(191 221)(192 240)(193 239)(194 238)(195 237)(196 236)(197 235)(198 234)(199 233)(200 232)

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

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

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

138 conjugacy classes

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

138 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 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 C2 C3 C6 C6 C6 C6 C6 D4 D4 D5 C4○D4 D10 D10 C3×D4 C3×D4 C3×D5 C5⋊D4 D20 C3×C4○D4 C6×D5 C6×D5 C4○D20 C3×C5⋊D4 C3×D20 C3×C4○D20 kernel C3×C20⋊7D4 C3×C4⋊Dic5 C3×D10⋊C4 C6×D20 C6×C5⋊D4 C22×C60 C20⋊7D4 C4⋊Dic5 D10⋊C4 C2×D20 C2×C5⋊D4 C22×C20 C60 C2×C30 C22×C12 C30 C2×C12 C22×C6 C20 C2×C10 C22×C4 C12 C2×C6 C10 C2×C4 C23 C6 C4 C22 C2 # reps 1 1 2 1 2 1 2 2 4 2 4 2 2 2 2 2 4 2 4 4 4 8 8 4 8 4 8 16 16 16

Matrix representation of C3×C207D4 in GL4(𝔽61) generated by

 13 0 0 0 0 13 0 0 0 0 1 0 0 0 0 1
,
 23 0 0 0 0 8 0 0 0 0 9 0 0 0 0 34
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 60 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[23,0,0,0,0,8,0,0,0,0,9,0,0,0,0,34],[0,1,0,0,1,0,0,0,0,0,0,60,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C3×C207D4 in GAP, Magma, Sage, TeX

C_3\times C_{20}\rtimes_7D_4
% in TeX

G:=Group("C3xC20:7D4");
// GroupNames label

G:=SmallGroup(480,723);
// by ID

G=gap.SmallGroup(480,723);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,344,590,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽