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## G = C5×Dic12order 240 = 24·3·5

### Direct product of C5 and Dic12

Aliases: C5×Dic12, C156Q16, C40.3S3, C24.1C10, C120.4C2, C30.30D4, C20.54D6, C10.15D12, C60.66C22, Dic6.1C10, C8.(C5×S3), C31(C5×Q16), C6.3(C5×D4), C2.5(C5×D12), C4.10(S3×C10), C12.10(C2×C10), (C5×Dic6).3C2, SmallGroup(240,53)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×Dic12
 Chief series C1 — C3 — C6 — C12 — C60 — C5×Dic6 — C5×Dic12
 Lower central C3 — C6 — C12 — C5×Dic12
 Upper central C1 — C10 — C20 — C40

Generators and relations for C5×Dic12
G = < a,b,c | a5=b24=1, c2=b12, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C5×Dic12
Regular action on 240 points
Generators in S240
(1 43 76 192 107)(2 44 77 169 108)(3 45 78 170 109)(4 46 79 171 110)(5 47 80 172 111)(6 48 81 173 112)(7 25 82 174 113)(8 26 83 175 114)(9 27 84 176 115)(10 28 85 177 116)(11 29 86 178 117)(12 30 87 179 118)(13 31 88 180 119)(14 32 89 181 120)(15 33 90 182 97)(16 34 91 183 98)(17 35 92 184 99)(18 36 93 185 100)(19 37 94 186 101)(20 38 95 187 102)(21 39 96 188 103)(22 40 73 189 104)(23 41 74 190 105)(24 42 75 191 106)(49 202 139 221 145)(50 203 140 222 146)(51 204 141 223 147)(52 205 142 224 148)(53 206 143 225 149)(54 207 144 226 150)(55 208 121 227 151)(56 209 122 228 152)(57 210 123 229 153)(58 211 124 230 154)(59 212 125 231 155)(60 213 126 232 156)(61 214 127 233 157)(62 215 128 234 158)(63 216 129 235 159)(64 193 130 236 160)(65 194 131 237 161)(66 195 132 238 162)(67 196 133 239 163)(68 197 134 240 164)(69 198 135 217 165)(70 199 136 218 166)(71 200 137 219 167)(72 201 138 220 168)
(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)
(1 145 13 157)(2 168 14 156)(3 167 15 155)(4 166 16 154)(5 165 17 153)(6 164 18 152)(7 163 19 151)(8 162 20 150)(9 161 21 149)(10 160 22 148)(11 159 23 147)(12 158 24 146)(25 67 37 55)(26 66 38 54)(27 65 39 53)(28 64 40 52)(29 63 41 51)(30 62 42 50)(31 61 43 49)(32 60 44 72)(33 59 45 71)(34 58 46 70)(35 57 47 69)(36 56 48 68)(73 205 85 193)(74 204 86 216)(75 203 87 215)(76 202 88 214)(77 201 89 213)(78 200 90 212)(79 199 91 211)(80 198 92 210)(81 197 93 209)(82 196 94 208)(83 195 95 207)(84 194 96 206)(97 231 109 219)(98 230 110 218)(99 229 111 217)(100 228 112 240)(101 227 113 239)(102 226 114 238)(103 225 115 237)(104 224 116 236)(105 223 117 235)(106 222 118 234)(107 221 119 233)(108 220 120 232)(121 174 133 186)(122 173 134 185)(123 172 135 184)(124 171 136 183)(125 170 137 182)(126 169 138 181)(127 192 139 180)(128 191 140 179)(129 190 141 178)(130 189 142 177)(131 188 143 176)(132 187 144 175)

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

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

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

C5×Dic12 is a maximal subgroup of
C40.D6  Dic12⋊D5  C15⋊Q32  C5⋊Dic24  C24.2D10  Dic10.D6  D120⋊C2  D405S3  D30.3D4  C5×S3×Q16

75 conjugacy classes

 class 1 2 3 4A 4B 4C 5A 5B 5C 5D 6 8A 8B 10A 10B 10C 10D 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 20E ··· 20L 24A 24B 24C 24D 30A 30B 30C 30D 40A ··· 40H 60A ··· 60H 120A ··· 120P order 1 2 3 4 4 4 5 5 5 5 6 8 8 10 10 10 10 12 12 15 15 15 15 20 20 20 20 20 ··· 20 24 24 24 24 30 30 30 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 2 2 12 12 1 1 1 1 2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 2 2 12 ··· 12 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

75 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - image C1 C2 C2 C5 C10 C10 S3 D4 D6 Q16 D12 C5×S3 C5×D4 Dic12 S3×C10 C5×Q16 C5×D12 C5×Dic12 kernel C5×Dic12 C120 C5×Dic6 Dic12 C24 Dic6 C40 C30 C20 C15 C10 C8 C6 C5 C4 C3 C2 C1 # reps 1 1 2 4 4 8 1 1 1 2 2 4 4 4 4 8 8 16

Matrix representation of C5×Dic12 in GL2(𝔽241) generated by

 205 0 0 205
,
 9 114 127 136
,
 91 125 34 150
G:=sub<GL(2,GF(241))| [205,0,0,205],[9,127,114,136],[91,34,125,150] >;

C5×Dic12 in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_{12}
% in TeX

G:=Group("C5xDic12");
// GroupNames label

G:=SmallGroup(240,53);
// by ID

G=gap.SmallGroup(240,53);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-3,240,265,367,1443,69,5765]);
// Polycyclic

G:=Group<a,b,c|a^5=b^24=1,c^2=b^12,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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