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G = C10×Dic6order 240 = 24·3·5

Direct product of C10 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C10×Dic6
 Chief series C1 — C3 — C6 — C30 — C5×Dic3 — C10×Dic3 — C10×Dic6
 Lower central C3 — C6 — C10×Dic6
 Upper central C1 — C2×C10 — C2×C20

Generators and relations for C10×Dic6
G = < a,b,c | a10=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 120 in 76 conjugacy classes, 54 normal (18 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, C20, C20, C2×C10, Dic6, C2×Dic3, C2×C12, C30, C30, C2×C20, C2×C20, C5×Q8, C2×Dic6, C5×Dic3, C60, C2×C30, Q8×C10, C5×Dic6, C10×Dic3, C2×C60, C10×Dic6
Quotients: C1, C2, C22, C5, S3, Q8, C23, C10, D6, C2×Q8, C2×C10, Dic6, C22×S3, C5×S3, C5×Q8, C22×C10, C2×Dic6, S3×C10, Q8×C10, C5×Dic6, S3×C2×C10, C10×Dic6

Smallest permutation representation of C10×Dic6
Regular action on 240 points
Generators in S240
(1 184 136 146 30 163 178 51 238 219)(2 185 137 147 31 164 179 52 239 220)(3 186 138 148 32 165 180 53 240 221)(4 187 139 149 33 166 169 54 229 222)(5 188 140 150 34 167 170 55 230 223)(6 189 141 151 35 168 171 56 231 224)(7 190 142 152 36 157 172 57 232 225)(8 191 143 153 25 158 173 58 233 226)(9 192 144 154 26 159 174 59 234 227)(10 181 133 155 27 160 175 60 235 228)(11 182 134 156 28 161 176 49 236 217)(12 183 135 145 29 162 177 50 237 218)(13 205 91 198 127 114 66 40 84 108)(14 206 92 199 128 115 67 41 73 97)(15 207 93 200 129 116 68 42 74 98)(16 208 94 201 130 117 69 43 75 99)(17 209 95 202 131 118 70 44 76 100)(18 210 96 203 132 119 71 45 77 101)(19 211 85 204 121 120 72 46 78 102)(20 212 86 193 122 109 61 47 79 103)(21 213 87 194 123 110 62 48 80 104)(22 214 88 195 124 111 63 37 81 105)(23 215 89 196 125 112 64 38 82 106)(24 216 90 197 126 113 65 39 83 107)
(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 44 7 38)(2 43 8 37)(3 42 9 48)(4 41 10 47)(5 40 11 46)(6 39 12 45)(13 156 19 150)(14 155 20 149)(15 154 21 148)(16 153 22 147)(17 152 23 146)(18 151 24 145)(25 214 31 208)(26 213 32 207)(27 212 33 206)(28 211 34 205)(29 210 35 216)(30 209 36 215)(49 121 55 127)(50 132 56 126)(51 131 57 125)(52 130 58 124)(53 129 59 123)(54 128 60 122)(61 222 67 228)(62 221 68 227)(63 220 69 226)(64 219 70 225)(65 218 71 224)(66 217 72 223)(73 181 79 187)(74 192 80 186)(75 191 81 185)(76 190 82 184)(77 189 83 183)(78 188 84 182)(85 167 91 161)(86 166 92 160)(87 165 93 159)(88 164 94 158)(89 163 95 157)(90 162 96 168)(97 133 103 139)(98 144 104 138)(99 143 105 137)(100 142 106 136)(101 141 107 135)(102 140 108 134)(109 229 115 235)(110 240 116 234)(111 239 117 233)(112 238 118 232)(113 237 119 231)(114 236 120 230)(169 199 175 193)(170 198 176 204)(171 197 177 203)(172 196 178 202)(173 195 179 201)(174 194 180 200)

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

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

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

90 conjugacy classes

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

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - + + - image C1 C2 C2 C2 C5 C10 C10 C10 S3 Q8 D6 D6 Dic6 C5×S3 C5×Q8 S3×C10 S3×C10 C5×Dic6 kernel C10×Dic6 C5×Dic6 C10×Dic3 C2×C60 C2×Dic6 Dic6 C2×Dic3 C2×C12 C2×C20 C30 C20 C2×C10 C10 C2×C4 C6 C4 C22 C2 # reps 1 4 2 1 4 16 8 4 1 2 2 1 4 4 8 8 4 16

Matrix representation of C10×Dic6 in GL4(𝔽61) generated by

 27 0 0 0 0 27 0 0 0 0 52 0 0 0 0 52
,
 60 1 0 0 60 0 0 0 0 0 15 38 0 0 23 38
,
 48 22 0 0 9 13 0 0 0 0 12 20 0 0 8 49
G:=sub<GL(4,GF(61))| [27,0,0,0,0,27,0,0,0,0,52,0,0,0,0,52],[60,60,0,0,1,0,0,0,0,0,15,23,0,0,38,38],[48,9,0,0,22,13,0,0,0,0,12,8,0,0,20,49] >;

C10×Dic6 in GAP, Magma, Sage, TeX

C_{10}\times {\rm Dic}_6
% in TeX

G:=Group("C10xDic6");
// GroupNames label

G:=SmallGroup(240,165);
// by ID

G=gap.SmallGroup(240,165);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-3,240,794,194,5765]);
// Polycyclic

G:=Group<a,b,c|a^10=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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