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## G = C4×Dic15order 240 = 24·3·5

### Direct product of C4 and Dic15

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4×Dic15
G = < a,b,c | a4=b30=1, c2=b15, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 200 in 60 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, C2×C4, C10, C10, Dic3, C12, C2×C6, C15, C42, Dic5, C20, C2×C10, C2×Dic3, C2×C12, C30, C30, C2×Dic5, C2×C20, C4×Dic3, Dic15, C60, C2×C30, C4×Dic5, C2×Dic15, C2×C60, C4×Dic15
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, Dic3, D6, C42, Dic5, D10, C4×S3, C2×Dic3, D15, C4×D5, C2×Dic5, C4×Dic3, Dic15, D30, C4×Dic5, C4×D15, C2×Dic15, C4×Dic15

Smallest permutation representation of C4×Dic15
Regular action on 240 points
Generators in S240
(1 182 71 166)(2 183 72 167)(3 184 73 168)(4 185 74 169)(5 186 75 170)(6 187 76 171)(7 188 77 172)(8 189 78 173)(9 190 79 174)(10 191 80 175)(11 192 81 176)(12 193 82 177)(13 194 83 178)(14 195 84 179)(15 196 85 180)(16 197 86 151)(17 198 87 152)(18 199 88 153)(19 200 89 154)(20 201 90 155)(21 202 61 156)(22 203 62 157)(23 204 63 158)(24 205 64 159)(25 206 65 160)(26 207 66 161)(27 208 67 162)(28 209 68 163)(29 210 69 164)(30 181 70 165)(31 100 212 140)(32 101 213 141)(33 102 214 142)(34 103 215 143)(35 104 216 144)(36 105 217 145)(37 106 218 146)(38 107 219 147)(39 108 220 148)(40 109 221 149)(41 110 222 150)(42 111 223 121)(43 112 224 122)(44 113 225 123)(45 114 226 124)(46 115 227 125)(47 116 228 126)(48 117 229 127)(49 118 230 128)(50 119 231 129)(51 120 232 130)(52 91 233 131)(53 92 234 132)(54 93 235 133)(55 94 236 134)(56 95 237 135)(57 96 238 136)(58 97 239 137)(59 98 240 138)(60 99 211 139)
(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)
(1 232 16 217)(2 231 17 216)(3 230 18 215)(4 229 19 214)(5 228 20 213)(6 227 21 212)(7 226 22 211)(8 225 23 240)(9 224 24 239)(10 223 25 238)(11 222 26 237)(12 221 27 236)(13 220 28 235)(14 219 29 234)(15 218 30 233)(31 76 46 61)(32 75 47 90)(33 74 48 89)(34 73 49 88)(35 72 50 87)(36 71 51 86)(37 70 52 85)(38 69 53 84)(39 68 54 83)(40 67 55 82)(41 66 56 81)(42 65 57 80)(43 64 58 79)(44 63 59 78)(45 62 60 77)(91 180 106 165)(92 179 107 164)(93 178 108 163)(94 177 109 162)(95 176 110 161)(96 175 111 160)(97 174 112 159)(98 173 113 158)(99 172 114 157)(100 171 115 156)(101 170 116 155)(102 169 117 154)(103 168 118 153)(104 167 119 152)(105 166 120 151)(121 206 136 191)(122 205 137 190)(123 204 138 189)(124 203 139 188)(125 202 140 187)(126 201 141 186)(127 200 142 185)(128 199 143 184)(129 198 144 183)(130 197 145 182)(131 196 146 181)(132 195 147 210)(133 194 148 209)(134 193 149 208)(135 192 150 207)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E ··· 4L 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 1 1 1 1 15 ··· 15 2 2 2 2 2 2 ··· 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + - + + - + image C1 C2 C2 C4 C4 S3 D5 Dic3 D6 Dic5 D10 C4×S3 D15 C4×D5 Dic15 D30 C4×D15 kernel C4×Dic15 C2×Dic15 C2×C60 Dic15 C60 C2×C20 C2×C12 C20 C2×C10 C12 C2×C6 C10 C2×C4 C6 C4 C22 C2 # reps 1 2 1 8 4 1 2 2 1 4 2 4 4 8 8 4 16

Matrix representation of C4×Dic15 in GL3(𝔽61) generated by

 60 0 0 0 11 0 0 0 11
,
 60 0 0 0 45 0 0 0 19
,
 50 0 0 0 0 60 0 1 0
G:=sub<GL(3,GF(61))| [60,0,0,0,11,0,0,0,11],[60,0,0,0,45,0,0,0,19],[50,0,0,0,0,1,0,60,0] >;

C4×Dic15 in GAP, Magma, Sage, TeX

C_4\times {\rm Dic}_{15}
% in TeX

G:=Group("C4xDic15");
// GroupNames label

G:=SmallGroup(240,72);
// by ID

G=gap.SmallGroup(240,72);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,55,964,6917]);
// Polycyclic

G:=Group<a,b,c|a^4=b^30=1,c^2=b^15,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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