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## G = Dic3×C20order 240 = 24·3·5

### Direct product of C20 and Dic3

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic3×C20
G = < a,b,c | a20=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 88 in 60 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×4], C22, C5, C6, C6 [×2], C2×C4, C2×C4 [×2], C10, C10 [×2], Dic3 [×4], C12 [×2], C2×C6, C15, C42, C20 [×2], C20 [×4], C2×C10, C2×Dic3 [×2], C2×C12, C30, C30 [×2], C2×C20, C2×C20 [×2], C4×Dic3, C5×Dic3 [×4], C60 [×2], C2×C30, C4×C20, C10×Dic3 [×2], C2×C60, Dic3×C20
Quotients: C1, C2 [×3], C4 [×6], C22, C5, S3, C2×C4 [×3], C10 [×3], Dic3 [×2], D6, C42, C20 [×6], C2×C10, C4×S3 [×2], C2×Dic3, C5×S3, C2×C20 [×3], C4×Dic3, C5×Dic3 [×2], S3×C10, C4×C20, S3×C20 [×2], C10×Dic3, Dic3×C20

Smallest permutation representation of Dic3×C20
Regular action on 240 points
Generators in S240
(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)
(1 159 58 95 104 231)(2 160 59 96 105 232)(3 141 60 97 106 233)(4 142 41 98 107 234)(5 143 42 99 108 235)(6 144 43 100 109 236)(7 145 44 81 110 237)(8 146 45 82 111 238)(9 147 46 83 112 239)(10 148 47 84 113 240)(11 149 48 85 114 221)(12 150 49 86 115 222)(13 151 50 87 116 223)(14 152 51 88 117 224)(15 153 52 89 118 225)(16 154 53 90 119 226)(17 155 54 91 120 227)(18 156 55 92 101 228)(19 157 56 93 102 229)(20 158 57 94 103 230)(21 190 64 179 129 208)(22 191 65 180 130 209)(23 192 66 161 131 210)(24 193 67 162 132 211)(25 194 68 163 133 212)(26 195 69 164 134 213)(27 196 70 165 135 214)(28 197 71 166 136 215)(29 198 72 167 137 216)(30 199 73 168 138 217)(31 200 74 169 139 218)(32 181 75 170 140 219)(33 182 76 171 121 220)(34 183 77 172 122 201)(35 184 78 173 123 202)(36 185 79 174 124 203)(37 186 80 175 125 204)(38 187 61 176 126 205)(39 188 62 177 127 206)(40 189 63 178 128 207)
(1 218 95 74)(2 219 96 75)(3 220 97 76)(4 201 98 77)(5 202 99 78)(6 203 100 79)(7 204 81 80)(8 205 82 61)(9 206 83 62)(10 207 84 63)(11 208 85 64)(12 209 86 65)(13 210 87 66)(14 211 88 67)(15 212 89 68)(16 213 90 69)(17 214 91 70)(18 215 92 71)(19 216 93 72)(20 217 94 73)(21 48 179 221)(22 49 180 222)(23 50 161 223)(24 51 162 224)(25 52 163 225)(26 53 164 226)(27 54 165 227)(28 55 166 228)(29 56 167 229)(30 57 168 230)(31 58 169 231)(32 59 170 232)(33 60 171 233)(34 41 172 234)(35 42 173 235)(36 43 174 236)(37 44 175 237)(38 45 176 238)(39 46 177 239)(40 47 178 240)(101 197 156 136)(102 198 157 137)(103 199 158 138)(104 200 159 139)(105 181 160 140)(106 182 141 121)(107 183 142 122)(108 184 143 123)(109 185 144 124)(110 186 145 125)(111 187 146 126)(112 188 147 127)(113 189 148 128)(114 190 149 129)(115 191 150 130)(116 192 151 131)(117 193 152 132)(118 194 153 133)(119 195 154 134)(120 196 155 135)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E ··· 4L 5A 5B 5C 5D 6A 6B 6C 10A ··· 10L 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20P 20Q ··· 20AV 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 1 1 1 1 3 ··· 3 1 1 1 1 2 2 2 1 ··· 1 2 2 2 2 2 2 2 2 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C4 C5 C10 C10 C20 C20 S3 Dic3 D6 C4×S3 C5×S3 C5×Dic3 S3×C10 S3×C20 kernel Dic3×C20 C10×Dic3 C2×C60 C5×Dic3 C60 C4×Dic3 C2×Dic3 C2×C12 Dic3 C12 C2×C20 C20 C2×C10 C10 C2×C4 C4 C22 C2 # reps 1 2 1 8 4 4 8 4 32 16 1 2 1 4 4 8 4 16

Matrix representation of Dic3×C20 in GL4(𝔽61) generated by

 34 0 0 0 0 50 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 0 60 0 0 0 0 0 1 0 0 60 60
,
 50 0 0 0 0 50 0 0 0 0 38 23 0 0 46 23
G:=sub<GL(4,GF(61))| [34,0,0,0,0,50,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,0,60,0,0,1,60],[50,0,0,0,0,50,0,0,0,0,38,46,0,0,23,23] >;

Dic3×C20 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{20}
% in TeX

G:=Group("Dic3xC20");
// GroupNames label

G:=SmallGroup(240,56);
// by ID

G=gap.SmallGroup(240,56);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-3,120,247,5765]);
// Polycyclic

G:=Group<a,b,c|a^20=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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