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## G = C3×C20.55D4order 480 = 25·3·5

### Direct product of C3 and C20.55D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C20.55D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C60 — C6×C5⋊2C8 — C3×C20.55D4
 Lower central C5 — C10 — C3×C20.55D4
 Upper central C1 — C2×C12 — C22×C12

Generators and relations for C3×C20.55D4
G = < a,b,c,d | a3=b20=1, c4=b10, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b9, dcd-1=b15c3 >

Subgroups: 192 in 100 conjugacy classes, 58 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×2], C22 [×2], C5, C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×2], C23, C10 [×3], C10 [×2], C12 [×2], C12, C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C2×C8 [×2], C22×C4, C20 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×2], C24 [×2], C2×C12 [×2], C2×C12 [×2], C22×C6, C30 [×3], C30 [×2], C22⋊C8, C52C8 [×2], C2×C20 [×2], C2×C20 [×2], C22×C10, C2×C24 [×2], C22×C12, C60 [×2], C60, C2×C30, C2×C30 [×2], C2×C30 [×2], C2×C52C8 [×2], C22×C20, C3×C22⋊C8, C3×C52C8 [×2], C2×C60 [×2], C2×C60 [×2], C22×C30, C20.55D4, C6×C52C8 [×2], C22×C60, C3×C20.55D4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C8 [×2], C2×C4, D4 [×2], D5, C12 [×2], C2×C6, C22⋊C4, C2×C8, M4(2), Dic5 [×2], D10, C24 [×2], C2×C12, C3×D4 [×2], C3×D5, C22⋊C8, C52C8 [×2], C2×Dic5, C5⋊D4 [×2], C3×C22⋊C4, C2×C24, C3×M4(2), C3×Dic5 [×2], C6×D5, C2×C52C8, C4.Dic5, C23.D5, C3×C22⋊C8, C3×C52C8 [×2], C6×Dic5, C3×C5⋊D4 [×2], C20.55D4, C6×C52C8, C3×C4.Dic5, C3×C23.D5, C3×C20.55D4

Smallest permutation representation of C3×C20.55D4
On 240 points
Generators in S240
(1 239 93)(2 240 94)(3 221 95)(4 222 96)(5 223 97)(6 224 98)(7 225 99)(8 226 100)(9 227 81)(10 228 82)(11 229 83)(12 230 84)(13 231 85)(14 232 86)(15 233 87)(16 234 88)(17 235 89)(18 236 90)(19 237 91)(20 238 92)(21 67 183)(22 68 184)(23 69 185)(24 70 186)(25 71 187)(26 72 188)(27 73 189)(28 74 190)(29 75 191)(30 76 192)(31 77 193)(32 78 194)(33 79 195)(34 80 196)(35 61 197)(36 62 198)(37 63 199)(38 64 200)(39 65 181)(40 66 182)(41 105 127)(42 106 128)(43 107 129)(44 108 130)(45 109 131)(46 110 132)(47 111 133)(48 112 134)(49 113 135)(50 114 136)(51 115 137)(52 116 138)(53 117 139)(54 118 140)(55 119 121)(56 120 122)(57 101 123)(58 102 124)(59 103 125)(60 104 126)(141 202 171)(142 203 172)(143 204 173)(144 205 174)(145 206 175)(146 207 176)(147 208 177)(148 209 178)(149 210 179)(150 211 180)(151 212 161)(152 213 162)(153 214 163)(154 215 164)(155 216 165)(156 217 166)(157 218 167)(158 219 168)(159 220 169)(160 201 170)
(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)
(1 189 60 213 11 199 50 203)(2 198 41 202 12 188 51 212)(3 187 42 211 13 197 52 201)(4 196 43 220 14 186 53 210)(5 185 44 209 15 195 54 219)(6 194 45 218 16 184 55 208)(7 183 46 207 17 193 56 217)(8 192 47 216 18 182 57 206)(9 181 48 205 19 191 58 215)(10 190 49 214 20 200 59 204)(21 110 176 235 31 120 166 225)(22 119 177 224 32 109 167 234)(23 108 178 233 33 118 168 223)(24 117 179 222 34 107 169 232)(25 106 180 231 35 116 170 221)(26 115 161 240 36 105 171 230)(27 104 162 229 37 114 172 239)(28 113 163 238 38 103 173 228)(29 102 164 227 39 112 174 237)(30 111 165 236 40 101 175 226)(61 138 160 95 71 128 150 85)(62 127 141 84 72 137 151 94)(63 136 142 93 73 126 152 83)(64 125 143 82 74 135 153 92)(65 134 144 91 75 124 154 81)(66 123 145 100 76 133 155 90)(67 132 146 89 77 122 156 99)(68 121 147 98 78 131 157 88)(69 130 148 87 79 140 158 97)(70 139 149 96 80 129 159 86)
(1 208 6 213 11 218 16 203)(2 217 7 202 12 207 17 212)(3 206 8 211 13 216 18 201)(4 215 9 220 14 205 19 210)(5 204 10 209 15 214 20 219)(21 110 26 115 31 120 36 105)(22 119 27 104 32 109 37 114)(23 108 28 113 33 118 38 103)(24 117 29 102 34 107 39 112)(25 106 30 111 35 116 40 101)(41 183 46 188 51 193 56 198)(42 192 47 197 52 182 57 187)(43 181 48 186 53 191 58 196)(44 190 49 195 54 200 59 185)(45 199 50 184 55 189 60 194)(61 138 66 123 71 128 76 133)(62 127 67 132 72 137 77 122)(63 136 68 121 73 126 78 131)(64 125 69 130 74 135 79 140)(65 134 70 139 75 124 80 129)(81 159 86 144 91 149 96 154)(82 148 87 153 92 158 97 143)(83 157 88 142 93 147 98 152)(84 146 89 151 94 156 99 141)(85 155 90 160 95 145 100 150)(161 240 166 225 171 230 176 235)(162 229 167 234 172 239 177 224)(163 238 168 223 173 228 178 233)(164 227 169 232 174 237 179 222)(165 236 170 221 175 226 180 231)

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

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

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

156 conjugacy classes

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

156 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 D4 D5 M4(2) Dic5 D10 Dic5 C3×D4 C3×D5 C5⋊D4 C5⋊2C8 C3×M4(2) C3×Dic5 C6×D5 C3×Dic5 C4.Dic5 C3×C5⋊D4 C3×C5⋊2C8 C3×C4.Dic5 kernel C3×C20.55D4 C6×C5⋊2C8 C22×C60 C20.55D4 C2×C60 C22×C30 C2×C5⋊2C8 C22×C20 C2×C30 C2×C20 C22×C10 C2×C10 C60 C22×C12 C30 C2×C12 C2×C12 C22×C6 C20 C22×C4 C12 C2×C6 C10 C2×C4 C2×C4 C23 C6 C4 C22 C2 # reps 1 2 1 2 2 2 4 2 8 4 4 16 2 2 2 2 2 2 4 4 8 8 4 4 4 4 8 16 16 16

Matrix representation of C3×C20.55D4 in GL3(𝔽241) generated by

 1 0 0 0 225 0 0 0 225
,
 177 0 0 0 25 0 0 0 106
,
 30 0 0 0 0 1 0 177 0
,
 211 0 0 0 0 1 0 64 0
G:=sub<GL(3,GF(241))| [1,0,0,0,225,0,0,0,225],[177,0,0,0,25,0,0,0,106],[30,0,0,0,0,177,0,1,0],[211,0,0,0,0,64,0,1,0] >;

C3×C20.55D4 in GAP, Magma, Sage, TeX

C_3\times C_{20}._{55}D_4
% in TeX

G:=Group("C3xC20.55D4");
// GroupNames label

G:=SmallGroup(480,108);
// by ID

G=gap.SmallGroup(480,108);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=1,c^4=b^10,d^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^9,d*c*d^-1=b^15*c^3>;
// generators/relations

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