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## G = C12×Dic5order 240 = 24·3·5

### Direct product of C12 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C12×Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — C6×Dic5 — C12×Dic5
 Lower central C5 — C12×Dic5
 Upper central C1 — C2×C12

Generators and relations for C12×Dic5
G = < a,b,c | a12=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 116 in 60 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, C2×C4, C10, C10, C12, C12, C2×C6, C15, C42, Dic5, C20, C2×C10, C2×C12, C2×C12, C30, C30, C2×Dic5, C2×C20, C4×C12, C3×Dic5, C60, C2×C30, C4×Dic5, C6×Dic5, C2×C60, C12×Dic5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D5, C12, C2×C6, C42, Dic5, D10, C2×C12, C3×D5, C4×D5, C2×Dic5, C4×C12, C3×Dic5, C6×D5, C4×Dic5, D5×C12, C6×Dic5, C12×Dic5

Smallest permutation representation of C12×Dic5
Regular action on 240 points
Generators in S240
(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)
(1 179 156 184 111 210 121 51 194 217)(2 180 145 185 112 211 122 52 195 218)(3 169 146 186 113 212 123 53 196 219)(4 170 147 187 114 213 124 54 197 220)(5 171 148 188 115 214 125 55 198 221)(6 172 149 189 116 215 126 56 199 222)(7 173 150 190 117 216 127 57 200 223)(8 174 151 191 118 205 128 58 201 224)(9 175 152 192 119 206 129 59 202 225)(10 176 153 181 120 207 130 60 203 226)(11 177 154 182 109 208 131 49 204 227)(12 178 155 183 110 209 132 50 193 228)(13 30 108 232 46 88 133 77 64 168)(14 31 97 233 47 89 134 78 65 157)(15 32 98 234 48 90 135 79 66 158)(16 33 99 235 37 91 136 80 67 159)(17 34 100 236 38 92 137 81 68 160)(18 35 101 237 39 93 138 82 69 161)(19 36 102 238 40 94 139 83 70 162)(20 25 103 239 41 95 140 84 71 163)(21 26 104 240 42 96 141 73 72 164)(22 27 105 229 43 85 142 74 61 165)(23 28 106 230 44 86 143 75 62 166)(24 29 107 231 45 87 144 76 63 167)
(1 67 210 235)(2 68 211 236)(3 69 212 237)(4 70 213 238)(5 71 214 239)(6 72 215 240)(7 61 216 229)(8 62 205 230)(9 63 206 231)(10 64 207 232)(11 65 208 233)(12 66 209 234)(13 181 88 203)(14 182 89 204)(15 183 90 193)(16 184 91 194)(17 185 92 195)(18 186 93 196)(19 187 94 197)(20 188 95 198)(21 189 96 199)(22 190 85 200)(23 191 86 201)(24 192 87 202)(25 148 140 55)(26 149 141 56)(27 150 142 57)(28 151 143 58)(29 152 144 59)(30 153 133 60)(31 154 134 49)(32 155 135 50)(33 156 136 51)(34 145 137 52)(35 146 138 53)(36 147 139 54)(37 217 159 111)(38 218 160 112)(39 219 161 113)(40 220 162 114)(41 221 163 115)(42 222 164 116)(43 223 165 117)(44 224 166 118)(45 225 167 119)(46 226 168 120)(47 227 157 109)(48 228 158 110)(73 126 104 172)(74 127 105 173)(75 128 106 174)(76 129 107 175)(77 130 108 176)(78 131 97 177)(79 132 98 178)(80 121 99 179)(81 122 100 180)(82 123 101 169)(83 124 102 170)(84 125 103 171)

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

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

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

96 conjugacy classes

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

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 D5 Dic5 D10 C3×D5 C4×D5 C3×Dic5 C6×D5 D5×C12 kernel C12×Dic5 C6×Dic5 C2×C60 C4×Dic5 C3×Dic5 C60 C2×Dic5 C2×C20 Dic5 C20 C2×C12 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 2 1 2 8 4 4 2 16 8 2 4 2 4 8 8 4 16

Matrix representation of C12×Dic5 in GL3(𝔽61) generated by

 50 0 0 0 21 0 0 0 21
,
 60 0 0 0 20 0 0 2 58
,
 50 0 0 0 60 42 0 0 1
G:=sub<GL(3,GF(61))| [50,0,0,0,21,0,0,0,21],[60,0,0,0,20,2,0,0,58],[50,0,0,0,60,0,0,42,1] >;

C12×Dic5 in GAP, Magma, Sage, TeX

C_{12}\times {\rm Dic}_5
% in TeX

G:=Group("C12xDic5");
// GroupNames label

G:=SmallGroup(240,40);
// by ID

G=gap.SmallGroup(240,40);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,72,151,6917]);
// Polycyclic

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

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