metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C30)⋊6D4, (C2×C10)⋊4D12, C5⋊4(C12⋊7D4), C6.164(D4×D5), C15⋊22(C4⋊D4), D6⋊Dic5⋊34C2, (C3×Dic5)⋊16D4, C10.69(C2×D12), C30.246(C2×D4), D30⋊4C4⋊34C2, C23.45(S3×D5), Dic5⋊5(C3⋊D4), C3⋊4(Dic5⋊D4), C22⋊2(C5⋊D12), C30.Q8⋊38C2, (C22×C6).95D10, (C22×C10).60D6, C30.157(C4○D4), C10.87(C4○D12), C6.60(D4⋊2D5), (C2×C30).208C23, (C2×Dic5).195D6, (C2×Dic3).64D10, (C22×Dic5)⋊10S3, (C22×S3).29D10, (C22×C30).70C22, C2.30(Dic3.D10), (C6×Dic5).224C22, (C22×D15).68C22, (C2×Dic15).142C22, (C10×Dic3).121C22, (C2×C3⋊D4)⋊6D5, (C2×C6×Dic5)⋊7C2, (C10×C3⋊D4)⋊6C2, (C2×C6)⋊8(C5⋊D4), C6.23(C2×C5⋊D4), C2.44(D5×C3⋊D4), (C2×C15⋊7D4)⋊17C2, (C2×C5⋊D12)⋊13C2, C2.24(C2×C5⋊D12), C10.68(C2×C3⋊D4), C22.237(C2×S3×D5), (S3×C2×C10).54C22, (C2×C6).220(C22×D5), (C2×C10).220(C22×S3), SmallGroup(480,642)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C10)⋊4D12
G = < a,b,c,d | a2=b10=c12=d2=1, ab=ba, ac=ca, dad=ab5, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 1020 in 188 conjugacy classes, 56 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, D15, C30, C30, C4⋊D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, C4⋊Dic3, D6⋊C4, C2×D12, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, S3×C10, D30, C2×C30, C2×C30, C2×C30, C10.D4, D10⋊C4, C23.D5, C22×Dic5, C2×C5⋊D4, D4×C10, C12⋊7D4, C5⋊D12, C6×Dic5, C6×Dic5, C10×Dic3, C5×C3⋊D4, C2×Dic15, C15⋊7D4, S3×C2×C10, C22×D15, C22×C30, Dic5⋊D4, D6⋊Dic5, D30⋊4C4, C30.Q8, C2×C5⋊D12, C2×C6×Dic5, C10×C3⋊D4, C2×C15⋊7D4, (C2×C10)⋊4D12
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, D12, C3⋊D4, C22×S3, C4⋊D4, C5⋊D4, C22×D5, C2×D12, C4○D12, C2×C3⋊D4, S3×D5, D4×D5, D4⋊2D5, C2×C5⋊D4, C12⋊7D4, C5⋊D12, C2×S3×D5, Dic5⋊D4, Dic3.D10, C2×C5⋊D12, D5×C3⋊D4, (C2×C10)⋊4D12
►On 240 points
Generators in S240
(25 58)(26 59)(27 60)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(73 88)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)(81 96)(82 85)(83 86)(84 87)(109 210)(110 211)(111 212)(112 213)(113 214)(114 215)(115 216)(116 205)(117 206)(118 207)(119 208)(120 209)(121 175)(122 176)(123 177)(124 178)(125 179)(126 180)(127 169)(128 170)(129 171)(130 172)(131 173)(132 174)(181 220)(182 221)(183 222)(184 223)(185 224)(186 225)(187 226)(188 227)(189 228)(190 217)(191 218)(192 219)
(1 46 229 140 199 23 161 152 104 70)(2 71 105 153 162 24 200 141 230 47)(3 48 231 142 201 13 163 154 106 72)(4 61 107 155 164 14 202 143 232 37)(5 38 233 144 203 15 165 156 108 62)(6 63 97 145 166 16 204 133 234 39)(7 40 235 134 193 17 167 146 98 64)(8 65 99 147 168 18 194 135 236 41)(9 42 237 136 195 19 157 148 100 66)(10 67 101 149 158 20 196 137 238 43)(11 44 239 138 197 21 159 150 102 68)(12 69 103 151 160 22 198 139 240 45)(25 111 184 93 177 58 212 223 78 123)(26 124 79 224 213 59 178 94 185 112)(27 113 186 95 179 60 214 225 80 125)(28 126 81 226 215 49 180 96 187 114)(29 115 188 85 169 50 216 227 82 127)(30 128 83 228 205 51 170 86 189 116)(31 117 190 87 171 52 206 217 84 129)(32 130 73 218 207 53 172 88 191 118)(33 119 192 89 173 54 208 219 74 131)(34 132 75 220 209 55 174 90 181 120)(35 109 182 91 175 56 210 221 76 121)(36 122 77 222 211 57 176 92 183 110)
(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 87)(2 86)(3 85)(4 96)(5 95)(6 94)(7 93)(8 92)(9 91)(10 90)(11 89)(12 88)(13 82)(14 81)(15 80)(16 79)(17 78)(18 77)(19 76)(20 75)(21 74)(22 73)(23 84)(24 83)(25 134)(26 133)(27 144)(28 143)(29 142)(30 141)(31 140)(32 139)(33 138)(34 137)(35 136)(36 135)(37 187)(38 186)(39 185)(40 184)(41 183)(42 182)(43 181)(44 192)(45 191)(46 190)(47 189)(48 188)(49 107)(50 106)(51 105)(52 104)(53 103)(54 102)(55 101)(56 100)(57 99)(58 98)(59 97)(60 108)(61 180)(62 179)(63 178)(64 177)(65 176)(66 175)(67 174)(68 173)(69 172)(70 171)(71 170)(72 169)(109 237)(110 236)(111 235)(112 234)(113 233)(114 232)(115 231)(116 230)(117 229)(118 240)(119 239)(120 238)(121 195)(122 194)(123 193)(124 204)(125 203)(126 202)(127 201)(128 200)(129 199)(130 198)(131 197)(132 196)(145 213)(146 212)(147 211)(148 210)(149 209)(150 208)(151 207)(152 206)(153 205)(154 216)(155 215)(156 214)(157 221)(158 220)(159 219)(160 218)(161 217)(162 228)(163 227)(164 226)(165 225)(166 224)(167 223)(168 222)
G:=sub<Sym(240)| (25,58)(26,59)(27,60)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(81,96)(82,85)(83,86)(84,87)(109,210)(110,211)(111,212)(112,213)(113,214)(114,215)(115,216)(116,205)(117,206)(118,207)(119,208)(120,209)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,169)(128,170)(129,171)(130,172)(131,173)(132,174)(181,220)(182,221)(183,222)(184,223)(185,224)(186,225)(187,226)(188,227)(189,228)(190,217)(191,218)(192,219), (1,46,229,140,199,23,161,152,104,70)(2,71,105,153,162,24,200,141,230,47)(3,48,231,142,201,13,163,154,106,72)(4,61,107,155,164,14,202,143,232,37)(5,38,233,144,203,15,165,156,108,62)(6,63,97,145,166,16,204,133,234,39)(7,40,235,134,193,17,167,146,98,64)(8,65,99,147,168,18,194,135,236,41)(9,42,237,136,195,19,157,148,100,66)(10,67,101,149,158,20,196,137,238,43)(11,44,239,138,197,21,159,150,102,68)(12,69,103,151,160,22,198,139,240,45)(25,111,184,93,177,58,212,223,78,123)(26,124,79,224,213,59,178,94,185,112)(27,113,186,95,179,60,214,225,80,125)(28,126,81,226,215,49,180,96,187,114)(29,115,188,85,169,50,216,227,82,127)(30,128,83,228,205,51,170,86,189,116)(31,117,190,87,171,52,206,217,84,129)(32,130,73,218,207,53,172,88,191,118)(33,119,192,89,173,54,208,219,74,131)(34,132,75,220,209,55,174,90,181,120)(35,109,182,91,175,56,210,221,76,121)(36,122,77,222,211,57,176,92,183,110), (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), (1,87)(2,86)(3,85)(4,96)(5,95)(6,94)(7,93)(8,92)(9,91)(10,90)(11,89)(12,88)(13,82)(14,81)(15,80)(16,79)(17,78)(18,77)(19,76)(20,75)(21,74)(22,73)(23,84)(24,83)(25,134)(26,133)(27,144)(28,143)(29,142)(30,141)(31,140)(32,139)(33,138)(34,137)(35,136)(36,135)(37,187)(38,186)(39,185)(40,184)(41,183)(42,182)(43,181)(44,192)(45,191)(46,190)(47,189)(48,188)(49,107)(50,106)(51,105)(52,104)(53,103)(54,102)(55,101)(56,100)(57,99)(58,98)(59,97)(60,108)(61,180)(62,179)(63,178)(64,177)(65,176)(66,175)(67,174)(68,173)(69,172)(70,171)(71,170)(72,169)(109,237)(110,236)(111,235)(112,234)(113,233)(114,232)(115,231)(116,230)(117,229)(118,240)(119,239)(120,238)(121,195)(122,194)(123,193)(124,204)(125,203)(126,202)(127,201)(128,200)(129,199)(130,198)(131,197)(132,196)(145,213)(146,212)(147,211)(148,210)(149,209)(150,208)(151,207)(152,206)(153,205)(154,216)(155,215)(156,214)(157,221)(158,220)(159,219)(160,218)(161,217)(162,228)(163,227)(164,226)(165,225)(166,224)(167,223)(168,222)>;
G:=Group( (25,58)(26,59)(27,60)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(81,96)(82,85)(83,86)(84,87)(109,210)(110,211)(111,212)(112,213)(113,214)(114,215)(115,216)(116,205)(117,206)(118,207)(119,208)(120,209)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,169)(128,170)(129,171)(130,172)(131,173)(132,174)(181,220)(182,221)(183,222)(184,223)(185,224)(186,225)(187,226)(188,227)(189,228)(190,217)(191,218)(192,219), (1,46,229,140,199,23,161,152,104,70)(2,71,105,153,162,24,200,141,230,47)(3,48,231,142,201,13,163,154,106,72)(4,61,107,155,164,14,202,143,232,37)(5,38,233,144,203,15,165,156,108,62)(6,63,97,145,166,16,204,133,234,39)(7,40,235,134,193,17,167,146,98,64)(8,65,99,147,168,18,194,135,236,41)(9,42,237,136,195,19,157,148,100,66)(10,67,101,149,158,20,196,137,238,43)(11,44,239,138,197,21,159,150,102,68)(12,69,103,151,160,22,198,139,240,45)(25,111,184,93,177,58,212,223,78,123)(26,124,79,224,213,59,178,94,185,112)(27,113,186,95,179,60,214,225,80,125)(28,126,81,226,215,49,180,96,187,114)(29,115,188,85,169,50,216,227,82,127)(30,128,83,228,205,51,170,86,189,116)(31,117,190,87,171,52,206,217,84,129)(32,130,73,218,207,53,172,88,191,118)(33,119,192,89,173,54,208,219,74,131)(34,132,75,220,209,55,174,90,181,120)(35,109,182,91,175,56,210,221,76,121)(36,122,77,222,211,57,176,92,183,110), (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), (1,87)(2,86)(3,85)(4,96)(5,95)(6,94)(7,93)(8,92)(9,91)(10,90)(11,89)(12,88)(13,82)(14,81)(15,80)(16,79)(17,78)(18,77)(19,76)(20,75)(21,74)(22,73)(23,84)(24,83)(25,134)(26,133)(27,144)(28,143)(29,142)(30,141)(31,140)(32,139)(33,138)(34,137)(35,136)(36,135)(37,187)(38,186)(39,185)(40,184)(41,183)(42,182)(43,181)(44,192)(45,191)(46,190)(47,189)(48,188)(49,107)(50,106)(51,105)(52,104)(53,103)(54,102)(55,101)(56,100)(57,99)(58,98)(59,97)(60,108)(61,180)(62,179)(63,178)(64,177)(65,176)(66,175)(67,174)(68,173)(69,172)(70,171)(71,170)(72,169)(109,237)(110,236)(111,235)(112,234)(113,233)(114,232)(115,231)(116,230)(117,229)(118,240)(119,239)(120,238)(121,195)(122,194)(123,193)(124,204)(125,203)(126,202)(127,201)(128,200)(129,199)(130,198)(131,197)(132,196)(145,213)(146,212)(147,211)(148,210)(149,209)(150,208)(151,207)(152,206)(153,205)(154,216)(155,215)(156,214)(157,221)(158,220)(159,219)(160,218)(161,217)(162,228)(163,227)(164,226)(165,225)(166,224)(167,223)(168,222) );
G=PermutationGroup([[(25,58),(26,59),(27,60),(28,49),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,57),(73,88),(74,89),(75,90),(76,91),(77,92),(78,93),(79,94),(80,95),(81,96),(82,85),(83,86),(84,87),(109,210),(110,211),(111,212),(112,213),(113,214),(114,215),(115,216),(116,205),(117,206),(118,207),(119,208),(120,209),(121,175),(122,176),(123,177),(124,178),(125,179),(126,180),(127,169),(128,170),(129,171),(130,172),(131,173),(132,174),(181,220),(182,221),(183,222),(184,223),(185,224),(186,225),(187,226),(188,227),(189,228),(190,217),(191,218),(192,219)], [(1,46,229,140,199,23,161,152,104,70),(2,71,105,153,162,24,200,141,230,47),(3,48,231,142,201,13,163,154,106,72),(4,61,107,155,164,14,202,143,232,37),(5,38,233,144,203,15,165,156,108,62),(6,63,97,145,166,16,204,133,234,39),(7,40,235,134,193,17,167,146,98,64),(8,65,99,147,168,18,194,135,236,41),(9,42,237,136,195,19,157,148,100,66),(10,67,101,149,158,20,196,137,238,43),(11,44,239,138,197,21,159,150,102,68),(12,69,103,151,160,22,198,139,240,45),(25,111,184,93,177,58,212,223,78,123),(26,124,79,224,213,59,178,94,185,112),(27,113,186,95,179,60,214,225,80,125),(28,126,81,226,215,49,180,96,187,114),(29,115,188,85,169,50,216,227,82,127),(30,128,83,228,205,51,170,86,189,116),(31,117,190,87,171,52,206,217,84,129),(32,130,73,218,207,53,172,88,191,118),(33,119,192,89,173,54,208,219,74,131),(34,132,75,220,209,55,174,90,181,120),(35,109,182,91,175,56,210,221,76,121),(36,122,77,222,211,57,176,92,183,110)], [(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)], [(1,87),(2,86),(3,85),(4,96),(5,95),(6,94),(7,93),(8,92),(9,91),(10,90),(11,89),(12,88),(13,82),(14,81),(15,80),(16,79),(17,78),(18,77),(19,76),(20,75),(21,74),(22,73),(23,84),(24,83),(25,134),(26,133),(27,144),(28,143),(29,142),(30,141),(31,140),(32,139),(33,138),(34,137),(35,136),(36,135),(37,187),(38,186),(39,185),(40,184),(41,183),(42,182),(43,181),(44,192),(45,191),(46,190),(47,189),(48,188),(49,107),(50,106),(51,105),(52,104),(53,103),(54,102),(55,101),(56,100),(57,99),(58,98),(59,97),(60,108),(61,180),(62,179),(63,178),(64,177),(65,176),(66,175),(67,174),(68,173),(69,172),(70,171),(71,170),(72,169),(109,237),(110,236),(111,235),(112,234),(113,233),(114,232),(115,231),(116,230),(117,229),(118,240),(119,239),(120,238),(121,195),(122,194),(123,193),(124,204),(125,203),(126,202),(127,201),(128,200),(129,199),(130,198),(131,197),(132,196),(145,213),(146,212),(147,211),(148,210),(149,209),(150,208),(151,207),(152,206),(153,205),(154,216),(155,215),(156,214),(157,221),(158,220),(159,219),(160,218),(161,217),(162,228),(163,227),(164,226),(165,225),(166,224),(167,223),(168,222)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | ··· | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 12A | ··· | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 60 | 2 | 10 | 10 | 10 | 10 | 12 | 60 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 10 | ··· | 10 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C3⋊D4 | D12 | C5⋊D4 | C4○D12 | S3×D5 | D4×D5 | D4⋊2D5 | C5⋊D12 | C2×S3×D5 | Dic3.D10 | D5×C3⋊D4 |
| kernel | (C2×C10)⋊4D12 | D6⋊Dic5 | D30⋊4C4 | C30.Q8 | C2×C5⋊D12 | C2×C6×Dic5 | C10×C3⋊D4 | C2×C15⋊7D4 | C22×Dic5 | C3×Dic5 | C2×C30 | C2×C3⋊D4 | C2×Dic5 | C22×C10 | C30 | C2×Dic3 | C22×S3 | C22×C6 | Dic5 | C2×C10 | C2×C6 | C10 | C23 | C6 | C6 | C22 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of (C2×C10)⋊4D12 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 17 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 18 | 18 | 0 | 0 | 0 | 0 |
| 43 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 34 | 9 | 0 | 0 | 0 | 0 |
| 7 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 4 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 0 |
| 0 | 0 | 0 | 0 | 47 | 40 |
| 47 | 55 | 0 | 0 | 0 | 0 |
| 2 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 59 | 0 | 0 |
| 0 | 0 | 37 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 53 | 15 |
| 0 | 0 | 0 | 0 | 8 | 8 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,17,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,43,0,0,0,0,18,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[34,7,0,0,0,0,9,27,0,0,0,0,0,0,11,0,0,0,0,0,4,50,0,0,0,0,0,0,29,47,0,0,0,0,0,40],[47,2,0,0,0,0,55,14,0,0,0,0,0,0,21,37,0,0,0,0,59,40,0,0,0,0,0,0,53,8,0,0,0,0,15,8] >;
(C2×C10)⋊4D12 in GAP, Magma, Sage, TeX
(C_2\times C_{10})\rtimes_4D_{12} % in TeX
G:=Group("(C2xC10):4D12"); // GroupNames label
G:=SmallGroup(480,642);
// by ID
G=gap.SmallGroup(480,642);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,219,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^12=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^5,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations