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## G = C3×C4.12D20order 480 = 25·3·5

### Direct product of C3 and C4.12D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×C4.12D20
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C60 — C6×Dic10 — C3×C4.12D20
 Lower central C5 — C10 — C2×C10 — C3×C4.12D20
 Upper central C1 — C6 — C2×C12 — C3×M4(2)

Generators and relations for C3×C4.12D20
G = < a,b,c,d | a3=b20=1, c4=d2=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b5c3 >

Subgroups: 224 in 76 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C2, C3, C4 [×2], C4 [×2], C22, C5, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], C10, C10, C12 [×2], C12 [×2], C2×C6, C15, M4(2), M4(2), C2×Q8, Dic5 [×2], C20 [×2], C2×C10, C24 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C30, C30, C4.10D4, C52C8, C40, Dic10 [×2], C2×Dic5 [×2], C2×C20, C3×M4(2), C3×M4(2), C6×Q8, C3×Dic5 [×2], C60 [×2], C2×C30, C4.Dic5, C5×M4(2), C2×Dic10, C3×C4.10D4, C3×C52C8, C120, C3×Dic10 [×2], C6×Dic5 [×2], C2×C60, C4.12D20, C3×C4.Dic5, C15×M4(2), C6×Dic10, C3×C4.12D20
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], D5, C12 [×2], C2×C6, C22⋊C4, D10, C2×C12, C3×D4 [×2], C3×D5, C4.10D4, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C6×D5, D10⋊C4, C3×C4.10D4, D5×C12, C3×D20, C3×C5⋊D4, C4.12D20, C3×D10⋊C4, C3×C4.12D20

Smallest permutation representation of C3×C4.12D20
On 240 points
Generators in S240
(1 45 32)(2 46 33)(3 47 34)(4 48 35)(5 49 36)(6 50 37)(7 51 38)(8 52 39)(9 53 40)(10 54 21)(11 55 22)(12 56 23)(13 57 24)(14 58 25)(15 59 26)(16 60 27)(17 41 28)(18 42 29)(19 43 30)(20 44 31)(61 115 81)(62 116 82)(63 117 83)(64 118 84)(65 119 85)(66 120 86)(67 101 87)(68 102 88)(69 103 89)(70 104 90)(71 105 91)(72 106 92)(73 107 93)(74 108 94)(75 109 95)(76 110 96)(77 111 97)(78 112 98)(79 113 99)(80 114 100)(121 176 141)(122 177 142)(123 178 143)(124 179 144)(125 180 145)(126 161 146)(127 162 147)(128 163 148)(129 164 149)(130 165 150)(131 166 151)(132 167 152)(133 168 153)(134 169 154)(135 170 155)(136 171 156)(137 172 157)(138 173 158)(139 174 159)(140 175 160)(181 226 206)(182 227 207)(183 228 208)(184 229 209)(185 230 210)(186 231 211)(187 232 212)(188 233 213)(189 234 214)(190 235 215)(191 236 216)(192 237 217)(193 238 218)(194 239 219)(195 240 220)(196 221 201)(197 222 202)(198 223 203)(199 224 204)(200 225 205)
(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)
(1 191 6 186 11 181 16 196)(2 190 7 185 12 200 17 195)(3 189 8 184 13 199 18 194)(4 188 9 183 14 198 19 193)(5 187 10 182 15 197 20 192)(21 207 26 202 31 217 36 212)(22 206 27 201 32 216 37 211)(23 205 28 220 33 215 38 210)(24 204 29 219 34 214 39 209)(25 203 30 218 35 213 40 208)(41 240 46 235 51 230 56 225)(42 239 47 234 52 229 57 224)(43 238 48 233 53 228 58 223)(44 237 49 232 54 227 59 222)(45 236 50 231 55 226 60 221)(61 136 76 121 71 126 66 131)(62 135 77 140 72 125 67 130)(63 134 78 139 73 124 68 129)(64 133 79 138 74 123 69 128)(65 132 80 137 75 122 70 127)(81 156 96 141 91 146 86 151)(82 155 97 160 92 145 87 150)(83 154 98 159 93 144 88 149)(84 153 99 158 94 143 89 148)(85 152 100 157 95 142 90 147)(101 165 116 170 111 175 106 180)(102 164 117 169 112 174 107 179)(103 163 118 168 113 173 108 178)(104 162 119 167 114 172 109 177)(105 161 120 166 115 171 110 176)
(1 136 11 126)(2 135 12 125)(3 134 13 124)(4 133 14 123)(5 132 15 122)(6 131 16 121)(7 130 17 140)(8 129 18 139)(9 128 19 138)(10 127 20 137)(21 147 31 157)(22 146 32 156)(23 145 33 155)(24 144 34 154)(25 143 35 153)(26 142 36 152)(27 141 37 151)(28 160 38 150)(29 159 39 149)(30 158 40 148)(41 175 51 165)(42 174 52 164)(43 173 53 163)(44 172 54 162)(45 171 55 161)(46 170 56 180)(47 169 57 179)(48 168 58 178)(49 167 59 177)(50 166 60 176)(61 186 71 196)(62 185 72 195)(63 184 73 194)(64 183 74 193)(65 182 75 192)(66 181 76 191)(67 200 77 190)(68 199 78 189)(69 198 79 188)(70 197 80 187)(81 211 91 201)(82 210 92 220)(83 209 93 219)(84 208 94 218)(85 207 95 217)(86 206 96 216)(87 205 97 215)(88 204 98 214)(89 203 99 213)(90 202 100 212)(101 225 111 235)(102 224 112 234)(103 223 113 233)(104 222 114 232)(105 221 115 231)(106 240 116 230)(107 239 117 229)(108 238 118 228)(109 237 119 227)(110 236 120 226)

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

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

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

93 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 8A 8B 8C 8D 10A 10B 10C 10D 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 24A 24B 24C 24D 24E 24F 24G 24H 30A 30B 30C 30D 30E 30F 30G 30H 40A ··· 40H 60A ··· 60H 60I 60J 60K 60L 120A ··· 120P order 1 2 2 3 3 4 4 4 4 5 5 6 6 6 6 8 8 8 8 10 10 10 10 12 12 12 12 12 12 12 12 15 15 15 15 20 20 20 20 20 20 24 24 24 24 24 24 24 24 30 30 30 30 30 30 30 30 40 ··· 40 60 ··· 60 60 60 60 60 120 ··· 120 size 1 1 2 1 1 2 2 20 20 2 2 1 1 2 2 4 4 20 20 2 2 4 4 2 2 2 2 20 20 20 20 2 2 2 2 2 2 2 2 4 4 4 4 4 4 20 20 20 20 2 2 2 2 4 4 4 4 4 ··· 4 2 ··· 2 4 4 4 4 4 ··· 4

93 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 D5 D10 C3×D4 C3×D5 D20 C5⋊D4 C4×D5 C6×D5 C3×D20 C3×C5⋊D4 D5×C12 C4.10D4 C3×C4.10D4 C4.12D20 C3×C4.12D20 kernel C3×C4.12D20 C3×C4.Dic5 C15×M4(2) C6×Dic10 C4.12D20 C6×Dic5 C4.Dic5 C5×M4(2) C2×Dic10 C2×Dic5 C60 C3×M4(2) C2×C12 C20 M4(2) C12 C12 C2×C6 C2×C4 C4 C4 C22 C15 C5 C3 C1 # reps 1 1 1 1 2 4 2 2 2 8 2 2 2 4 4 4 4 4 4 8 8 8 1 2 4 8

Matrix representation of C3×C4.12D20 in GL4(𝔽241) generated by

 15 0 0 0 0 15 0 0 0 0 15 0 0 0 0 15
,
 238 197 0 0 44 163 0 0 0 0 41 197 0 0 163 119
,
 60 148 196 203 75 181 225 83 47 36 135 181 70 230 119 106
,
 60 148 0 0 75 181 0 0 0 0 106 60 0 0 122 135
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[238,44,0,0,197,163,0,0,0,0,41,163,0,0,197,119],[60,75,47,70,148,181,36,230,196,225,135,119,203,83,181,106],[60,75,0,0,148,181,0,0,0,0,106,122,0,0,60,135] >;

C3×C4.12D20 in GAP, Magma, Sage, TeX

C_3\times C_4._{12}D_{20}
% in TeX

G:=Group("C3xC4.12D20");
// GroupNames label

G:=SmallGroup(480,102);
// by ID

G=gap.SmallGroup(480,102);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,336,365,92,1683,136,1271,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=1,c^4=d^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^5*c^3>;
// generators/relations

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