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## G = (C2×C10).D12order 480 = 25·3·5

### 8th non-split extension by C2×C10 of D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — (C2×C10).D12
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — C30.Q8 — (C2×C10).D12
 Lower central C15 — C2×C30 — (C2×C10).D12
 Upper central C1 — C22 — C23

Generators and relations for (C2×C10).D12
G = < a,b,c,d | a2=b10=c12=1, d2=b5, ab=ba, cac-1=ab5, ad=da, cbc-1=dbd-1=b-1, dcd-1=c-1 >

Subgroups: 684 in 156 conjugacy classes, 52 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, C23, C23, C10, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C5×S3, C30, C30, C30, C22.D4, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, C22×C10, C4⋊Dic3, D6⋊C4, C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, S3×C10, C2×C30, C2×C30, C2×C30, C10.D4, C23.D5, C23.D5, C22×Dic5, D4×C10, C23.21D6, C6×Dic5, C10×Dic3, C5×C3⋊D4, C2×Dic15, C2×Dic15, S3×C2×C10, C22×C30, C23.18D10, D6⋊Dic5, C30.Q8, C3×C23.D5, C10×C3⋊D4, C22×Dic15, (C2×C10).D12
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, D12, C22×S3, C22.D4, C5⋊D4, C22×D5, C2×D12, D42S3, S3×D5, D42D5, C2×C5⋊D4, C23.21D6, C5⋊D12, C2×S3×D5, C23.18D10, C30.C23, C2×C5⋊D12, (C2×C10).D12

Smallest permutation representation of (C2×C10).D12
On 240 points
Generators in S240
(2 181)(4 183)(6 185)(8 187)(10 189)(12 191)(13 238)(15 240)(17 230)(19 232)(21 234)(23 236)(25 101)(27 103)(29 105)(31 107)(33 97)(35 99)(38 223)(40 225)(42 227)(44 217)(46 219)(48 221)(49 87)(51 89)(53 91)(55 93)(57 95)(59 85)(61 119)(63 109)(65 111)(67 113)(69 115)(71 117)(73 162)(75 164)(77 166)(79 168)(81 158)(83 160)(121 180)(123 170)(125 172)(127 174)(129 176)(131 178)(133 213)(135 215)(137 205)(139 207)(141 209)(143 211)(145 203)(147 193)(149 195)(151 197)(153 199)(155 201)
(1 118 22 138 52 192 72 235 206 90)(2 91 207 236 61 181 53 139 23 119)(3 120 24 140 54 182 62 237 208 92)(4 93 209 238 63 183 55 141 13 109)(5 110 14 142 56 184 64 239 210 94)(6 95 211 240 65 185 57 143 15 111)(7 112 16 144 58 186 66 229 212 96)(8 85 213 230 67 187 59 133 17 113)(9 114 18 134 60 188 68 231 214 86)(10 87 215 232 69 189 49 135 19 115)(11 116 20 136 50 190 70 233 216 88)(12 89 205 234 71 191 51 137 21 117)(25 158 129 40 197 101 81 176 225 151)(26 152 226 177 82 102 198 41 130 159)(27 160 131 42 199 103 83 178 227 153)(28 154 228 179 84 104 200 43 132 161)(29 162 121 44 201 105 73 180 217 155)(30 156 218 169 74 106 202 45 122 163)(31 164 123 46 203 107 75 170 219 145)(32 146 220 171 76 108 204 47 124 165)(33 166 125 48 193 97 77 172 221 147)(34 148 222 173 78 98 194 37 126 167)(35 168 127 38 195 99 79 174 223 149)(36 150 224 175 80 100 196 39 128 157)
(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)
(1 179 192 132)(2 178 181 131)(3 177 182 130)(4 176 183 129)(5 175 184 128)(6 174 185 127)(7 173 186 126)(8 172 187 125)(9 171 188 124)(10 170 189 123)(11 169 190 122)(12 180 191 121)(13 151 238 197)(14 150 239 196)(15 149 240 195)(16 148 229 194)(17 147 230 193)(18 146 231 204)(19 145 232 203)(20 156 233 202)(21 155 234 201)(22 154 235 200)(23 153 236 199)(24 152 237 198)(25 209 101 141)(26 208 102 140)(27 207 103 139)(28 206 104 138)(29 205 105 137)(30 216 106 136)(31 215 107 135)(32 214 108 134)(33 213 97 133)(34 212 98 144)(35 211 99 143)(36 210 100 142)(37 112 222 66)(38 111 223 65)(39 110 224 64)(40 109 225 63)(41 120 226 62)(42 119 227 61)(43 118 228 72)(44 117 217 71)(45 116 218 70)(46 115 219 69)(47 114 220 68)(48 113 221 67)(49 164 87 75)(50 163 88 74)(51 162 89 73)(52 161 90 84)(53 160 91 83)(54 159 92 82)(55 158 93 81)(56 157 94 80)(57 168 95 79)(58 167 96 78)(59 166 85 77)(60 165 86 76)

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 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 S3 D4 D5 D6 D6 C4○D4 D10 D10 D10 D12 C5⋊D4 D4⋊2S3 S3×D5 D4⋊2D5 C5⋊D12 C2×S3×D5 C30.C23 kernel (C2×C10).D12 D6⋊Dic5 C30.Q8 C3×C23.D5 C10×C3⋊D4 C22×Dic15 C23.D5 C2×C30 C2×C3⋊D4 C2×Dic5 C22×C10 C30 C2×Dic3 C22×S3 C22×C6 C2×C10 C2×C6 C10 C23 C6 C22 C22 C2 # reps 1 2 2 1 1 1 1 2 2 2 1 4 2 2 2 4 8 2 2 4 4 2 8

Matrix representation of (C2×C10).D12 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 20 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 44 0 0 0 0 18 43
,
 60 46 0 0 0 0 49 2 0 0 0 0 0 0 11 5 0 0 0 0 37 50 0 0 0 0 0 0 31 16 0 0 0 0 39 30
,
 35 12 0 0 0 0 20 26 0 0 0 0 0 0 50 0 0 0 0 0 0 50 0 0 0 0 0 0 1 0 0 0 0 0 19 60

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,20,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,18,0,0,0,0,44,43],[60,49,0,0,0,0,46,2,0,0,0,0,0,0,11,37,0,0,0,0,5,50,0,0,0,0,0,0,31,39,0,0,0,0,16,30],[35,20,0,0,0,0,12,26,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,1,19,0,0,0,0,0,60] >;

(C2×C10).D12 in GAP, Magma, Sage, TeX

(C_2\times C_{10}).D_{12}
% in TeX

G:=Group("(C2xC10).D12");
// GroupNames label

G:=SmallGroup(480,619);
// by ID

G=gap.SmallGroup(480,619);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,120,422,219,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^12=1,d^2=b^5,a*b=b*a,c*a*c^-1=a*b^5,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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