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## G = C2×Dic30order 240 = 24·3·5

### Direct product of C2 and Dic30

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×Dic30
 Chief series C1 — C5 — C15 — C30 — Dic15 — C2×Dic15 — C2×Dic30
 Lower central C15 — C30 — C2×Dic30
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×Dic30
G = < a,b,c | a2=b60=1, c2=b30, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 296 in 76 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, C2×C4, Q8, C10, C10, Dic3, C12, C2×C6, C15, C2×Q8, Dic5, C20, C2×C10, Dic6, C2×Dic3, C2×C12, C30, C30, Dic10, C2×Dic5, C2×C20, C2×Dic6, Dic15, C60, C2×C30, C2×Dic10, Dic30, C2×Dic15, C2×C60, C2×Dic30
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, D10, Dic6, C22×S3, D15, Dic10, C22×D5, C2×Dic6, D30, C2×Dic10, Dic30, C22×D15, C2×Dic30

Smallest permutation representation of C2×Dic30
Regular action on 240 points
Generators in S240
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 81)(16 82)(17 83)(18 84)(19 85)(20 86)(21 87)(22 88)(23 89)(24 90)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 97)(32 98)(33 99)(34 100)(35 101)(36 102)(37 103)(38 104)(39 105)(40 106)(41 107)(42 108)(43 109)(44 110)(45 111)(46 112)(47 113)(48 114)(49 115)(50 116)(51 117)(52 118)(53 119)(54 120)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(121 188)(122 189)(123 190)(124 191)(125 192)(126 193)(127 194)(128 195)(129 196)(130 197)(131 198)(132 199)(133 200)(134 201)(135 202)(136 203)(137 204)(138 205)(139 206)(140 207)(141 208)(142 209)(143 210)(144 211)(145 212)(146 213)(147 214)(148 215)(149 216)(150 217)(151 218)(152 219)(153 220)(154 221)(155 222)(156 223)(157 224)(158 225)(159 226)(160 227)(161 228)(162 229)(163 230)(164 231)(165 232)(166 233)(167 234)(168 235)(169 236)(170 237)(171 238)(172 239)(173 240)(174 181)(175 182)(176 183)(177 184)(178 185)(179 186)(180 187)
(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)
(1 238 31 208)(2 237 32 207)(3 236 33 206)(4 235 34 205)(5 234 35 204)(6 233 36 203)(7 232 37 202)(8 231 38 201)(9 230 39 200)(10 229 40 199)(11 228 41 198)(12 227 42 197)(13 226 43 196)(14 225 44 195)(15 224 45 194)(16 223 46 193)(17 222 47 192)(18 221 48 191)(19 220 49 190)(20 219 50 189)(21 218 51 188)(22 217 52 187)(23 216 53 186)(24 215 54 185)(25 214 55 184)(26 213 56 183)(27 212 57 182)(28 211 58 181)(29 210 59 240)(30 209 60 239)(61 177 91 147)(62 176 92 146)(63 175 93 145)(64 174 94 144)(65 173 95 143)(66 172 96 142)(67 171 97 141)(68 170 98 140)(69 169 99 139)(70 168 100 138)(71 167 101 137)(72 166 102 136)(73 165 103 135)(74 164 104 134)(75 163 105 133)(76 162 106 132)(77 161 107 131)(78 160 108 130)(79 159 109 129)(80 158 110 128)(81 157 111 127)(82 156 112 126)(83 155 113 125)(84 154 114 124)(85 153 115 123)(86 152 116 122)(87 151 117 121)(88 150 118 180)(89 149 119 179)(90 148 120 178)

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + + + + + - + - + + - image C1 C2 C2 C2 S3 Q8 D5 D6 D6 D10 D10 Dic6 D15 Dic10 D30 D30 Dic30 kernel C2×Dic30 Dic30 C2×Dic15 C2×C60 C2×C20 C30 C2×C12 C20 C2×C10 C12 C2×C6 C10 C2×C4 C6 C4 C22 C2 # reps 1 4 2 1 1 2 2 2 1 4 2 4 4 8 8 4 16

Matrix representation of C2×Dic30 in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 0 60 0 0 1 18 0 0 0 0 55 12 0 0 49 34
,
 18 18 0 0 60 43 0 0 0 0 11 0 0 0 57 50
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,60,18,0,0,0,0,55,49,0,0,12,34],[18,60,0,0,18,43,0,0,0,0,11,57,0,0,0,50] >;

C2×Dic30 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_{30}
% in TeX

G:=Group("C2xDic30");
// GroupNames label

G:=SmallGroup(240,175);
// by ID

G=gap.SmallGroup(240,175);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,218,50,964,6917]);
// Polycyclic

G:=Group<a,b,c|a^2=b^60=1,c^2=b^30,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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