direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C20.23D4, C60.133D4, (C6×Q8)⋊11D5, (Q8×C10)⋊8C6, (Q8×C30)⋊11C2, (C4×Dic5)⋊7C6, C10.58(C6×D4), C20.23(C3×D4), (C2×D20).10C6, (C6×D20).21C2, C30.415(C2×D4), D10⋊C4⋊16C6, (C12×Dic5)⋊19C2, (C2×C12).245D10, C12.79(C5⋊D4), C15⋊26(C4.4D4), C30.267(C4○D4), (C2×C30).376C23, (C2×C60).300C22, C6.55(Q8⋊2D5), (C6×Dic5).255C22, (C2×Q8)⋊6(C3×D5), C5⋊4(C3×C4.4D4), (C2×C4).57(C6×D5), C2.22(C6×C5⋊D4), C4.11(C3×C5⋊D4), C22.65(D5×C2×C6), (C2×C20).64(C2×C6), C10.37(C3×C4○D4), C6.143(C2×C5⋊D4), C2.9(C3×Q8⋊2D5), (D5×C2×C6).85C22, (C3×D10⋊C4)⋊38C2, (C2×C10).59(C22×C6), (C2×Dic5).43(C2×C6), (C22×D5).15(C2×C6), (C2×C6).372(C22×D5), SmallGroup(480,740)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C20.23D4
G = < a,b,c,d | a3=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b9, dbd=b-1, dcd=b10c-1 >
Subgroups: 544 in 152 conjugacy classes, 66 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C3×D5, C30, C30, C4.4D4, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C4×C12, C3×C22⋊C4, C6×D4, C6×Q8, C3×Dic5, C60, C60, C6×D5, C2×C30, C4×Dic5, D10⋊C4, C2×D20, Q8×C10, C3×C4.4D4, C3×D20, C6×Dic5, C2×C60, C2×C60, Q8×C15, D5×C2×C6, C20.23D4, C12×Dic5, C3×D10⋊C4, C6×D20, Q8×C30, C3×C20.23D4
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, C4○D4, D10, C3×D4, C22×C6, C3×D5, C4.4D4, C5⋊D4, C22×D5, C6×D4, C3×C4○D4, C6×D5, Q8⋊2D5, C2×C5⋊D4, C3×C4.4D4, C3×C5⋊D4, D5×C2×C6, C20.23D4, C3×Q8⋊2D5, C6×C5⋊D4, C3×C20.23D4
►On 240 points
Generators in S240
(1 47 34)(2 48 35)(3 49 36)(4 50 37)(5 51 38)(6 52 39)(7 53 40)(8 54 21)(9 55 22)(10 56 23)(11 57 24)(12 58 25)(13 59 26)(14 60 27)(15 41 28)(16 42 29)(17 43 30)(18 44 31)(19 45 32)(20 46 33)(61 127 158)(62 128 159)(63 129 160)(64 130 141)(65 131 142)(66 132 143)(67 133 144)(68 134 145)(69 135 146)(70 136 147)(71 137 148)(72 138 149)(73 139 150)(74 140 151)(75 121 152)(76 122 153)(77 123 154)(78 124 155)(79 125 156)(80 126 157)(81 191 103)(82 192 104)(83 193 105)(84 194 106)(85 195 107)(86 196 108)(87 197 109)(88 198 110)(89 199 111)(90 200 112)(91 181 113)(92 182 114)(93 183 115)(94 184 116)(95 185 117)(96 186 118)(97 187 119)(98 188 120)(99 189 101)(100 190 102)(161 238 215)(162 239 216)(163 240 217)(164 221 218)(165 222 219)(166 223 220)(167 224 201)(168 225 202)(169 226 203)(170 227 204)(171 228 205)(172 229 206)(173 230 207)(174 231 208)(175 232 209)(176 233 210)(177 234 211)(178 235 212)(179 236 213)(180 237 214)
(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)
(1 177 153 98)(2 166 154 87)(3 175 155 96)(4 164 156 85)(5 173 157 94)(6 162 158 83)(7 171 159 92)(8 180 160 81)(9 169 141 90)(10 178 142 99)(11 167 143 88)(12 176 144 97)(13 165 145 86)(14 174 146 95)(15 163 147 84)(16 172 148 93)(17 161 149 82)(18 170 150 91)(19 179 151 100)(20 168 152 89)(21 214 129 103)(22 203 130 112)(23 212 131 101)(24 201 132 110)(25 210 133 119)(26 219 134 108)(27 208 135 117)(28 217 136 106)(29 206 137 115)(30 215 138 104)(31 204 139 113)(32 213 140 102)(33 202 121 111)(34 211 122 120)(35 220 123 109)(36 209 124 118)(37 218 125 107)(38 207 126 116)(39 216 127 105)(40 205 128 114)(41 240 70 194)(42 229 71 183)(43 238 72 192)(44 227 73 181)(45 236 74 190)(46 225 75 199)(47 234 76 188)(48 223 77 197)(49 232 78 186)(50 221 79 195)(51 230 80 184)(52 239 61 193)(53 228 62 182)(54 237 63 191)(55 226 64 200)(56 235 65 189)(57 224 66 198)(58 233 67 187)(59 222 68 196)(60 231 69 185)
(1 158)(2 157)(3 156)(4 155)(5 154)(6 153)(7 152)(8 151)(9 150)(10 149)(11 148)(12 147)(13 146)(14 145)(15 144)(16 143)(17 142)(18 141)(19 160)(20 159)(21 140)(22 139)(23 138)(24 137)(25 136)(26 135)(27 134)(28 133)(29 132)(30 131)(31 130)(32 129)(33 128)(34 127)(35 126)(36 125)(37 124)(38 123)(39 122)(40 121)(41 67)(42 66)(43 65)(44 64)(45 63)(46 62)(47 61)(48 80)(49 79)(50 78)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)(57 71)(58 70)(59 69)(60 68)(81 90)(82 89)(83 88)(84 87)(85 86)(91 100)(92 99)(93 98)(94 97)(95 96)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(115 120)(116 119)(117 118)(161 168)(162 167)(163 166)(164 165)(169 180)(170 179)(171 178)(172 177)(173 176)(174 175)(181 190)(182 189)(183 188)(184 187)(185 186)(191 200)(192 199)(193 198)(194 197)(195 196)(201 216)(202 215)(203 214)(204 213)(205 212)(206 211)(207 210)(208 209)(217 220)(218 219)(221 222)(223 240)(224 239)(225 238)(226 237)(227 236)(228 235)(229 234)(230 233)(231 232)
G:=sub<Sym(240)| (1,47,34)(2,48,35)(3,49,36)(4,50,37)(5,51,38)(6,52,39)(7,53,40)(8,54,21)(9,55,22)(10,56,23)(11,57,24)(12,58,25)(13,59,26)(14,60,27)(15,41,28)(16,42,29)(17,43,30)(18,44,31)(19,45,32)(20,46,33)(61,127,158)(62,128,159)(63,129,160)(64,130,141)(65,131,142)(66,132,143)(67,133,144)(68,134,145)(69,135,146)(70,136,147)(71,137,148)(72,138,149)(73,139,150)(74,140,151)(75,121,152)(76,122,153)(77,123,154)(78,124,155)(79,125,156)(80,126,157)(81,191,103)(82,192,104)(83,193,105)(84,194,106)(85,195,107)(86,196,108)(87,197,109)(88,198,110)(89,199,111)(90,200,112)(91,181,113)(92,182,114)(93,183,115)(94,184,116)(95,185,117)(96,186,118)(97,187,119)(98,188,120)(99,189,101)(100,190,102)(161,238,215)(162,239,216)(163,240,217)(164,221,218)(165,222,219)(166,223,220)(167,224,201)(168,225,202)(169,226,203)(170,227,204)(171,228,205)(172,229,206)(173,230,207)(174,231,208)(175,232,209)(176,233,210)(177,234,211)(178,235,212)(179,236,213)(180,237,214), (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), (1,177,153,98)(2,166,154,87)(3,175,155,96)(4,164,156,85)(5,173,157,94)(6,162,158,83)(7,171,159,92)(8,180,160,81)(9,169,141,90)(10,178,142,99)(11,167,143,88)(12,176,144,97)(13,165,145,86)(14,174,146,95)(15,163,147,84)(16,172,148,93)(17,161,149,82)(18,170,150,91)(19,179,151,100)(20,168,152,89)(21,214,129,103)(22,203,130,112)(23,212,131,101)(24,201,132,110)(25,210,133,119)(26,219,134,108)(27,208,135,117)(28,217,136,106)(29,206,137,115)(30,215,138,104)(31,204,139,113)(32,213,140,102)(33,202,121,111)(34,211,122,120)(35,220,123,109)(36,209,124,118)(37,218,125,107)(38,207,126,116)(39,216,127,105)(40,205,128,114)(41,240,70,194)(42,229,71,183)(43,238,72,192)(44,227,73,181)(45,236,74,190)(46,225,75,199)(47,234,76,188)(48,223,77,197)(49,232,78,186)(50,221,79,195)(51,230,80,184)(52,239,61,193)(53,228,62,182)(54,237,63,191)(55,226,64,200)(56,235,65,189)(57,224,66,198)(58,233,67,187)(59,222,68,196)(60,231,69,185), (1,158)(2,157)(3,156)(4,155)(5,154)(6,153)(7,152)(8,151)(9,150)(10,149)(11,148)(12,147)(13,146)(14,145)(15,144)(16,143)(17,142)(18,141)(19,160)(20,159)(21,140)(22,139)(23,138)(24,137)(25,136)(26,135)(27,134)(28,133)(29,132)(30,131)(31,130)(32,129)(33,128)(34,127)(35,126)(36,125)(37,124)(38,123)(39,122)(40,121)(41,67)(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,80)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68)(81,90)(82,89)(83,88)(84,87)(85,86)(91,100)(92,99)(93,98)(94,97)(95,96)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(115,120)(116,119)(117,118)(161,168)(162,167)(163,166)(164,165)(169,180)(170,179)(171,178)(172,177)(173,176)(174,175)(181,190)(182,189)(183,188)(184,187)(185,186)(191,200)(192,199)(193,198)(194,197)(195,196)(201,216)(202,215)(203,214)(204,213)(205,212)(206,211)(207,210)(208,209)(217,220)(218,219)(221,222)(223,240)(224,239)(225,238)(226,237)(227,236)(228,235)(229,234)(230,233)(231,232)>;
G:=Group( (1,47,34)(2,48,35)(3,49,36)(4,50,37)(5,51,38)(6,52,39)(7,53,40)(8,54,21)(9,55,22)(10,56,23)(11,57,24)(12,58,25)(13,59,26)(14,60,27)(15,41,28)(16,42,29)(17,43,30)(18,44,31)(19,45,32)(20,46,33)(61,127,158)(62,128,159)(63,129,160)(64,130,141)(65,131,142)(66,132,143)(67,133,144)(68,134,145)(69,135,146)(70,136,147)(71,137,148)(72,138,149)(73,139,150)(74,140,151)(75,121,152)(76,122,153)(77,123,154)(78,124,155)(79,125,156)(80,126,157)(81,191,103)(82,192,104)(83,193,105)(84,194,106)(85,195,107)(86,196,108)(87,197,109)(88,198,110)(89,199,111)(90,200,112)(91,181,113)(92,182,114)(93,183,115)(94,184,116)(95,185,117)(96,186,118)(97,187,119)(98,188,120)(99,189,101)(100,190,102)(161,238,215)(162,239,216)(163,240,217)(164,221,218)(165,222,219)(166,223,220)(167,224,201)(168,225,202)(169,226,203)(170,227,204)(171,228,205)(172,229,206)(173,230,207)(174,231,208)(175,232,209)(176,233,210)(177,234,211)(178,235,212)(179,236,213)(180,237,214), (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), (1,177,153,98)(2,166,154,87)(3,175,155,96)(4,164,156,85)(5,173,157,94)(6,162,158,83)(7,171,159,92)(8,180,160,81)(9,169,141,90)(10,178,142,99)(11,167,143,88)(12,176,144,97)(13,165,145,86)(14,174,146,95)(15,163,147,84)(16,172,148,93)(17,161,149,82)(18,170,150,91)(19,179,151,100)(20,168,152,89)(21,214,129,103)(22,203,130,112)(23,212,131,101)(24,201,132,110)(25,210,133,119)(26,219,134,108)(27,208,135,117)(28,217,136,106)(29,206,137,115)(30,215,138,104)(31,204,139,113)(32,213,140,102)(33,202,121,111)(34,211,122,120)(35,220,123,109)(36,209,124,118)(37,218,125,107)(38,207,126,116)(39,216,127,105)(40,205,128,114)(41,240,70,194)(42,229,71,183)(43,238,72,192)(44,227,73,181)(45,236,74,190)(46,225,75,199)(47,234,76,188)(48,223,77,197)(49,232,78,186)(50,221,79,195)(51,230,80,184)(52,239,61,193)(53,228,62,182)(54,237,63,191)(55,226,64,200)(56,235,65,189)(57,224,66,198)(58,233,67,187)(59,222,68,196)(60,231,69,185), (1,158)(2,157)(3,156)(4,155)(5,154)(6,153)(7,152)(8,151)(9,150)(10,149)(11,148)(12,147)(13,146)(14,145)(15,144)(16,143)(17,142)(18,141)(19,160)(20,159)(21,140)(22,139)(23,138)(24,137)(25,136)(26,135)(27,134)(28,133)(29,132)(30,131)(31,130)(32,129)(33,128)(34,127)(35,126)(36,125)(37,124)(38,123)(39,122)(40,121)(41,67)(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,80)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68)(81,90)(82,89)(83,88)(84,87)(85,86)(91,100)(92,99)(93,98)(94,97)(95,96)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(115,120)(116,119)(117,118)(161,168)(162,167)(163,166)(164,165)(169,180)(170,179)(171,178)(172,177)(173,176)(174,175)(181,190)(182,189)(183,188)(184,187)(185,186)(191,200)(192,199)(193,198)(194,197)(195,196)(201,216)(202,215)(203,214)(204,213)(205,212)(206,211)(207,210)(208,209)(217,220)(218,219)(221,222)(223,240)(224,239)(225,238)(226,237)(227,236)(228,235)(229,234)(230,233)(231,232) );
G=PermutationGroup([[(1,47,34),(2,48,35),(3,49,36),(4,50,37),(5,51,38),(6,52,39),(7,53,40),(8,54,21),(9,55,22),(10,56,23),(11,57,24),(12,58,25),(13,59,26),(14,60,27),(15,41,28),(16,42,29),(17,43,30),(18,44,31),(19,45,32),(20,46,33),(61,127,158),(62,128,159),(63,129,160),(64,130,141),(65,131,142),(66,132,143),(67,133,144),(68,134,145),(69,135,146),(70,136,147),(71,137,148),(72,138,149),(73,139,150),(74,140,151),(75,121,152),(76,122,153),(77,123,154),(78,124,155),(79,125,156),(80,126,157),(81,191,103),(82,192,104),(83,193,105),(84,194,106),(85,195,107),(86,196,108),(87,197,109),(88,198,110),(89,199,111),(90,200,112),(91,181,113),(92,182,114),(93,183,115),(94,184,116),(95,185,117),(96,186,118),(97,187,119),(98,188,120),(99,189,101),(100,190,102),(161,238,215),(162,239,216),(163,240,217),(164,221,218),(165,222,219),(166,223,220),(167,224,201),(168,225,202),(169,226,203),(170,227,204),(171,228,205),(172,229,206),(173,230,207),(174,231,208),(175,232,209),(176,233,210),(177,234,211),(178,235,212),(179,236,213),(180,237,214)], [(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)], [(1,177,153,98),(2,166,154,87),(3,175,155,96),(4,164,156,85),(5,173,157,94),(6,162,158,83),(7,171,159,92),(8,180,160,81),(9,169,141,90),(10,178,142,99),(11,167,143,88),(12,176,144,97),(13,165,145,86),(14,174,146,95),(15,163,147,84),(16,172,148,93),(17,161,149,82),(18,170,150,91),(19,179,151,100),(20,168,152,89),(21,214,129,103),(22,203,130,112),(23,212,131,101),(24,201,132,110),(25,210,133,119),(26,219,134,108),(27,208,135,117),(28,217,136,106),(29,206,137,115),(30,215,138,104),(31,204,139,113),(32,213,140,102),(33,202,121,111),(34,211,122,120),(35,220,123,109),(36,209,124,118),(37,218,125,107),(38,207,126,116),(39,216,127,105),(40,205,128,114),(41,240,70,194),(42,229,71,183),(43,238,72,192),(44,227,73,181),(45,236,74,190),(46,225,75,199),(47,234,76,188),(48,223,77,197),(49,232,78,186),(50,221,79,195),(51,230,80,184),(52,239,61,193),(53,228,62,182),(54,237,63,191),(55,226,64,200),(56,235,65,189),(57,224,66,198),(58,233,67,187),(59,222,68,196),(60,231,69,185)], [(1,158),(2,157),(3,156),(4,155),(5,154),(6,153),(7,152),(8,151),(9,150),(10,149),(11,148),(12,147),(13,146),(14,145),(15,144),(16,143),(17,142),(18,141),(19,160),(20,159),(21,140),(22,139),(23,138),(24,137),(25,136),(26,135),(27,134),(28,133),(29,132),(30,131),(31,130),(32,129),(33,128),(34,127),(35,126),(36,125),(37,124),(38,123),(39,122),(40,121),(41,67),(42,66),(43,65),(44,64),(45,63),(46,62),(47,61),(48,80),(49,79),(50,78),(51,77),(52,76),(53,75),(54,74),(55,73),(56,72),(57,71),(58,70),(59,69),(60,68),(81,90),(82,89),(83,88),(84,87),(85,86),(91,100),(92,99),(93,98),(94,97),(95,96),(101,114),(102,113),(103,112),(104,111),(105,110),(106,109),(107,108),(115,120),(116,119),(117,118),(161,168),(162,167),(163,166),(164,165),(169,180),(170,179),(171,178),(172,177),(173,176),(174,175),(181,190),(182,189),(183,188),(184,187),(185,186),(191,200),(192,199),(193,198),(194,197),(195,196),(201,216),(202,215),(203,214),(204,213),(205,212),(206,211),(207,210),(208,209),(217,220),(218,219),(221,222),(223,240),(224,239),(225,238),(226,237),(227,236),(228,235),(229,234),(230,233),(231,232)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | ··· | 12P | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | D5 | C4○D4 | D10 | C3×D4 | C3×D5 | C5⋊D4 | C3×C4○D4 | C6×D5 | C3×C5⋊D4 | Q8⋊2D5 | C3×Q8⋊2D5 |
| kernel | C3×C20.23D4 | C12×Dic5 | C3×D10⋊C4 | C6×D20 | Q8×C30 | C20.23D4 | C4×Dic5 | D10⋊C4 | C2×D20 | Q8×C10 | C60 | C6×Q8 | C30 | C2×C12 | C20 | C2×Q8 | C12 | C10 | C2×C4 | C4 | C6 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 4 | 6 | 4 | 4 | 8 | 8 | 12 | 16 | 4 | 8 |
Matrix representation of C3×C20.23D4 ►in GL4(𝔽61) generated by
| 47 | 0 | 0 | 0 |
| 0 | 47 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 60 | 1 | 0 | 0 |
| 16 | 44 | 0 | 0 |
| 0 | 0 | 50 | 39 |
| 0 | 0 | 0 | 11 |
| 22 | 31 | 0 | 0 |
| 6 | 39 | 0 | 0 |
| 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 50 |
| 44 | 60 | 0 | 0 |
| 44 | 17 | 0 | 0 |
| 0 | 0 | 50 | 39 |
| 0 | 0 | 11 | 11 |
G:=sub<GL(4,GF(61))| [47,0,0,0,0,47,0,0,0,0,13,0,0,0,0,13],[60,16,0,0,1,44,0,0,0,0,50,0,0,0,39,11],[22,6,0,0,31,39,0,0,0,0,50,0,0,0,0,50],[44,44,0,0,60,17,0,0,0,0,50,11,0,0,39,11] >;
C3×C20.23D4 in GAP, Magma, Sage, TeX
C_3\times C_{20}._{23}D_4 % in TeX
G:=Group("C3xC20.23D4"); // GroupNames label
G:=SmallGroup(480,740);
// by ID
G=gap.SmallGroup(480,740);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,344,590,555,268,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^9,d*b*d=b^-1,d*c*d=b^10*c^-1>;
// generators/relations