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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, C3, C4, C4, C22, C5, C6, C6, C2×C4, C2×C4, C10, C10, Dic3, C12, C2×C6, C15, C42, C20, C20, C2×C10, C2×Dic3, C2×C12, C30, C30, C2×C20, C2×C20, C4×Dic3, C5×Dic3, C60, C2×C30, C4×C20, C10×Dic3, C2×C60, Dic3×C20
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C10, Dic3, D6, C42, C20, C2×C10, C4×S3, C2×Dic3, C5×S3, C2×C20, C4×Dic3, C5×Dic3, S3×C10, C4×C20, S3×C20, 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)
(1 161 58 235 92 213)(2 162 59 236 93 214)(3 163 60 237 94 215)(4 164 41 238 95 216)(5 165 42 239 96 217)(6 166 43 240 97 218)(7 167 44 221 98 219)(8 168 45 222 99 220)(9 169 46 223 100 201)(10 170 47 224 81 202)(11 171 48 225 82 203)(12 172 49 226 83 204)(13 173 50 227 84 205)(14 174 51 228 85 206)(15 175 52 229 86 207)(16 176 53 230 87 208)(17 177 54 231 88 209)(18 178 55 232 89 210)(19 179 56 233 90 211)(20 180 57 234 91 212)(21 150 66 138 184 119)(22 151 67 139 185 120)(23 152 68 140 186 101)(24 153 69 121 187 102)(25 154 70 122 188 103)(26 155 71 123 189 104)(27 156 72 124 190 105)(28 157 73 125 191 106)(29 158 74 126 192 107)(30 159 75 127 193 108)(31 160 76 128 194 109)(32 141 77 129 195 110)(33 142 78 130 196 111)(34 143 79 131 197 112)(35 144 80 132 198 113)(36 145 61 133 199 114)(37 146 62 134 200 115)(38 147 63 135 181 116)(39 148 64 136 182 117)(40 149 65 137 183 118)
(1 136 235 39)(2 137 236 40)(3 138 237 21)(4 139 238 22)(5 140 239 23)(6 121 240 24)(7 122 221 25)(8 123 222 26)(9 124 223 27)(10 125 224 28)(11 126 225 29)(12 127 226 30)(13 128 227 31)(14 129 228 32)(15 130 229 33)(16 131 230 34)(17 132 231 35)(18 133 232 36)(19 134 233 37)(20 135 234 38)(41 151 216 185)(42 152 217 186)(43 153 218 187)(44 154 219 188)(45 155 220 189)(46 156 201 190)(47 157 202 191)(48 158 203 192)(49 159 204 193)(50 160 205 194)(51 141 206 195)(52 142 207 196)(53 143 208 197)(54 144 209 198)(55 145 210 199)(56 146 211 200)(57 147 212 181)(58 148 213 182)(59 149 214 183)(60 150 215 184)(61 89 114 178)(62 90 115 179)(63 91 116 180)(64 92 117 161)(65 93 118 162)(66 94 119 163)(67 95 120 164)(68 96 101 165)(69 97 102 166)(70 98 103 167)(71 99 104 168)(72 100 105 169)(73 81 106 170)(74 82 107 171)(75 83 108 172)(76 84 109 173)(77 85 110 174)(78 86 111 175)(79 87 112 176)(80 88 113 177)

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

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

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

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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