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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 [×2], C3, C4 [×2], C4 [×4], C22, C5, C6, C6 [×2], C2×C4, C2×C4 [×2], Q8 [×4], C10, C10 [×2], Dic3 [×4], C12 [×2], C2×C6, C15, C2×Q8, C20 [×2], C20 [×4], C2×C10, Dic6 [×4], C2×Dic3 [×2], C2×C12, C30, C30 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×4], C2×Dic6, C5×Dic3 [×4], C60 [×2], C2×C30, Q8×C10, C5×Dic6 [×4], C10×Dic3 [×2], C2×C60, C10×Dic6
Quotients: C1, C2 [×7], C22 [×7], C5, S3, Q8 [×2], C23, C10 [×7], D6 [×3], C2×Q8, C2×C10 [×7], Dic6 [×2], C22×S3, C5×S3, C5×Q8 [×2], C22×C10, C2×Dic6, S3×C10 [×3], Q8×C10, C5×Dic6 [×2], S3×C2×C10, C10×Dic6

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

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

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

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

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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