direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C80⋊C2, C80⋊5C6, C48⋊7D5, C240⋊11C2, D10.1C24, C24.81D10, C15⋊11M5(2), Dic5.1C24, C120.99C22, C16⋊3(C3×D5), C5⋊2C16⋊4C6, (C8×D5).2C6, (C6×D5).3C8, C8.20(C6×D5), C6.17(C8×D5), C2.3(D5×C24), C5⋊3(C3×M5(2)), C5⋊2C8.2C12, C30.49(C2×C8), C40.20(C2×C6), (D5×C12).8C4, (C4×D5).2C12, (D5×C24).9C2, C4.17(D5×C12), C12.87(C4×D5), C60.209(C2×C4), C10.11(C2×C24), C20.43(C2×C12), (C3×Dic5).3C8, (C3×C5⋊2C8).7C4, (C3×C5⋊2C16)⋊11C2, SmallGroup(480,76)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C80⋊C2
G = < a,b,c | a3=b80=c2=1, ab=ba, ac=ca, cbc=b9 >
Smallest permutation representation of C3×C80⋊C2
►On 240 points
Generators in S240
(1 117 176)(2 118 177)(3 119 178)(4 120 179)(5 121 180)(6 122 181)(7 123 182)(8 124 183)(9 125 184)(10 126 185)(11 127 186)(12 128 187)(13 129 188)(14 130 189)(15 131 190)(16 132 191)(17 133 192)(18 134 193)(19 135 194)(20 136 195)(21 137 196)(22 138 197)(23 139 198)(24 140 199)(25 141 200)(26 142 201)(27 143 202)(28 144 203)(29 145 204)(30 146 205)(31 147 206)(32 148 207)(33 149 208)(34 150 209)(35 151 210)(36 152 211)(37 153 212)(38 154 213)(39 155 214)(40 156 215)(41 157 216)(42 158 217)(43 159 218)(44 160 219)(45 81 220)(46 82 221)(47 83 222)(48 84 223)(49 85 224)(50 86 225)(51 87 226)(52 88 227)(53 89 228)(54 90 229)(55 91 230)(56 92 231)(57 93 232)(58 94 233)(59 95 234)(60 96 235)(61 97 236)(62 98 237)(63 99 238)(64 100 239)(65 101 240)(66 102 161)(67 103 162)(68 104 163)(69 105 164)(70 106 165)(71 107 166)(72 108 167)(73 109 168)(74 110 169)(75 111 170)(76 112 171)(77 113 172)(78 114 173)(79 115 174)(80 116 175)
(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)
(2 10)(3 19)(4 28)(5 37)(6 46)(7 55)(8 64)(9 73)(12 20)(13 29)(14 38)(15 47)(16 56)(17 65)(18 74)(22 30)(23 39)(24 48)(25 57)(26 66)(27 75)(32 40)(33 49)(34 58)(35 67)(36 76)(42 50)(43 59)(44 68)(45 77)(52 60)(53 69)(54 78)(62 70)(63 79)(72 80)(81 113)(82 122)(83 131)(84 140)(85 149)(86 158)(88 96)(89 105)(90 114)(91 123)(92 132)(93 141)(94 150)(95 159)(98 106)(99 115)(100 124)(101 133)(102 142)(103 151)(104 160)(108 116)(109 125)(110 134)(111 143)(112 152)(118 126)(119 135)(120 144)(121 153)(128 136)(129 145)(130 154)(138 146)(139 155)(148 156)(161 201)(162 210)(163 219)(164 228)(165 237)(167 175)(168 184)(169 193)(170 202)(171 211)(172 220)(173 229)(174 238)(177 185)(178 194)(179 203)(180 212)(181 221)(182 230)(183 239)(187 195)(188 204)(189 213)(190 222)(191 231)(192 240)(197 205)(198 214)(199 223)(200 232)(207 215)(208 224)(209 233)(217 225)(218 234)(227 235)
G:=sub<Sym(240)| (1,117,176)(2,118,177)(3,119,178)(4,120,179)(5,121,180)(6,122,181)(7,123,182)(8,124,183)(9,125,184)(10,126,185)(11,127,186)(12,128,187)(13,129,188)(14,130,189)(15,131,190)(16,132,191)(17,133,192)(18,134,193)(19,135,194)(20,136,195)(21,137,196)(22,138,197)(23,139,198)(24,140,199)(25,141,200)(26,142,201)(27,143,202)(28,144,203)(29,145,204)(30,146,205)(31,147,206)(32,148,207)(33,149,208)(34,150,209)(35,151,210)(36,152,211)(37,153,212)(38,154,213)(39,155,214)(40,156,215)(41,157,216)(42,158,217)(43,159,218)(44,160,219)(45,81,220)(46,82,221)(47,83,222)(48,84,223)(49,85,224)(50,86,225)(51,87,226)(52,88,227)(53,89,228)(54,90,229)(55,91,230)(56,92,231)(57,93,232)(58,94,233)(59,95,234)(60,96,235)(61,97,236)(62,98,237)(63,99,238)(64,100,239)(65,101,240)(66,102,161)(67,103,162)(68,104,163)(69,105,164)(70,106,165)(71,107,166)(72,108,167)(73,109,168)(74,110,169)(75,111,170)(76,112,171)(77,113,172)(78,114,173)(79,115,174)(80,116,175), (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), (2,10)(3,19)(4,28)(5,37)(6,46)(7,55)(8,64)(9,73)(12,20)(13,29)(14,38)(15,47)(16,56)(17,65)(18,74)(22,30)(23,39)(24,48)(25,57)(26,66)(27,75)(32,40)(33,49)(34,58)(35,67)(36,76)(42,50)(43,59)(44,68)(45,77)(52,60)(53,69)(54,78)(62,70)(63,79)(72,80)(81,113)(82,122)(83,131)(84,140)(85,149)(86,158)(88,96)(89,105)(90,114)(91,123)(92,132)(93,141)(94,150)(95,159)(98,106)(99,115)(100,124)(101,133)(102,142)(103,151)(104,160)(108,116)(109,125)(110,134)(111,143)(112,152)(118,126)(119,135)(120,144)(121,153)(128,136)(129,145)(130,154)(138,146)(139,155)(148,156)(161,201)(162,210)(163,219)(164,228)(165,237)(167,175)(168,184)(169,193)(170,202)(171,211)(172,220)(173,229)(174,238)(177,185)(178,194)(179,203)(180,212)(181,221)(182,230)(183,239)(187,195)(188,204)(189,213)(190,222)(191,231)(192,240)(197,205)(198,214)(199,223)(200,232)(207,215)(208,224)(209,233)(217,225)(218,234)(227,235)>;
G:=Group( (1,117,176)(2,118,177)(3,119,178)(4,120,179)(5,121,180)(6,122,181)(7,123,182)(8,124,183)(9,125,184)(10,126,185)(11,127,186)(12,128,187)(13,129,188)(14,130,189)(15,131,190)(16,132,191)(17,133,192)(18,134,193)(19,135,194)(20,136,195)(21,137,196)(22,138,197)(23,139,198)(24,140,199)(25,141,200)(26,142,201)(27,143,202)(28,144,203)(29,145,204)(30,146,205)(31,147,206)(32,148,207)(33,149,208)(34,150,209)(35,151,210)(36,152,211)(37,153,212)(38,154,213)(39,155,214)(40,156,215)(41,157,216)(42,158,217)(43,159,218)(44,160,219)(45,81,220)(46,82,221)(47,83,222)(48,84,223)(49,85,224)(50,86,225)(51,87,226)(52,88,227)(53,89,228)(54,90,229)(55,91,230)(56,92,231)(57,93,232)(58,94,233)(59,95,234)(60,96,235)(61,97,236)(62,98,237)(63,99,238)(64,100,239)(65,101,240)(66,102,161)(67,103,162)(68,104,163)(69,105,164)(70,106,165)(71,107,166)(72,108,167)(73,109,168)(74,110,169)(75,111,170)(76,112,171)(77,113,172)(78,114,173)(79,115,174)(80,116,175), (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), (2,10)(3,19)(4,28)(5,37)(6,46)(7,55)(8,64)(9,73)(12,20)(13,29)(14,38)(15,47)(16,56)(17,65)(18,74)(22,30)(23,39)(24,48)(25,57)(26,66)(27,75)(32,40)(33,49)(34,58)(35,67)(36,76)(42,50)(43,59)(44,68)(45,77)(52,60)(53,69)(54,78)(62,70)(63,79)(72,80)(81,113)(82,122)(83,131)(84,140)(85,149)(86,158)(88,96)(89,105)(90,114)(91,123)(92,132)(93,141)(94,150)(95,159)(98,106)(99,115)(100,124)(101,133)(102,142)(103,151)(104,160)(108,116)(109,125)(110,134)(111,143)(112,152)(118,126)(119,135)(120,144)(121,153)(128,136)(129,145)(130,154)(138,146)(139,155)(148,156)(161,201)(162,210)(163,219)(164,228)(165,237)(167,175)(168,184)(169,193)(170,202)(171,211)(172,220)(173,229)(174,238)(177,185)(178,194)(179,203)(180,212)(181,221)(182,230)(183,239)(187,195)(188,204)(189,213)(190,222)(191,231)(192,240)(197,205)(198,214)(199,223)(200,232)(207,215)(208,224)(209,233)(217,225)(218,234)(227,235) );
G=PermutationGroup([[(1,117,176),(2,118,177),(3,119,178),(4,120,179),(5,121,180),(6,122,181),(7,123,182),(8,124,183),(9,125,184),(10,126,185),(11,127,186),(12,128,187),(13,129,188),(14,130,189),(15,131,190),(16,132,191),(17,133,192),(18,134,193),(19,135,194),(20,136,195),(21,137,196),(22,138,197),(23,139,198),(24,140,199),(25,141,200),(26,142,201),(27,143,202),(28,144,203),(29,145,204),(30,146,205),(31,147,206),(32,148,207),(33,149,208),(34,150,209),(35,151,210),(36,152,211),(37,153,212),(38,154,213),(39,155,214),(40,156,215),(41,157,216),(42,158,217),(43,159,218),(44,160,219),(45,81,220),(46,82,221),(47,83,222),(48,84,223),(49,85,224),(50,86,225),(51,87,226),(52,88,227),(53,89,228),(54,90,229),(55,91,230),(56,92,231),(57,93,232),(58,94,233),(59,95,234),(60,96,235),(61,97,236),(62,98,237),(63,99,238),(64,100,239),(65,101,240),(66,102,161),(67,103,162),(68,104,163),(69,105,164),(70,106,165),(71,107,166),(72,108,167),(73,109,168),(74,110,169),(75,111,170),(76,112,171),(77,113,172),(78,114,173),(79,115,174),(80,116,175)], [(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)], [(2,10),(3,19),(4,28),(5,37),(6,46),(7,55),(8,64),(9,73),(12,20),(13,29),(14,38),(15,47),(16,56),(17,65),(18,74),(22,30),(23,39),(24,48),(25,57),(26,66),(27,75),(32,40),(33,49),(34,58),(35,67),(36,76),(42,50),(43,59),(44,68),(45,77),(52,60),(53,69),(54,78),(62,70),(63,79),(72,80),(81,113),(82,122),(83,131),(84,140),(85,149),(86,158),(88,96),(89,105),(90,114),(91,123),(92,132),(93,141),(94,150),(95,159),(98,106),(99,115),(100,124),(101,133),(102,142),(103,151),(104,160),(108,116),(109,125),(110,134),(111,143),(112,152),(118,126),(119,135),(120,144),(121,153),(128,136),(129,145),(130,154),(138,146),(139,155),(148,156),(161,201),(162,210),(163,219),(164,228),(165,237),(167,175),(168,184),(169,193),(170,202),(171,211),(172,220),(173,229),(174,238),(177,185),(178,194),(179,203),(180,212),(181,221),(182,230),(183,239),(187,195),(188,204),(189,213),(190,222),(191,231),(192,240),(197,205),(198,214),(199,223),(200,232),(207,215),(208,224),(209,233),(217,225),(218,234),(227,235)]])
156 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 15C | 15D | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 20C | 20D | 24A | ··· | 24H | 24I | 24J | 24K | 24L | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 48A | ··· | 48H | 48I | ··· | 48P | 60A | ··· | 60H | 80A | ··· | 80P | 120A | ··· | 120P | 240A | ··· | 240AF |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 48 | ··· | 48 | 48 | ··· | 48 | 60 | ··· | 60 | 80 | ··· | 80 | 120 | ··· | 120 | 240 | ··· | 240 |
| size | 1 | 1 | 10 | 1 | 1 | 1 | 1 | 10 | 2 | 2 | 1 | 1 | 10 | 10 | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
156 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C8 | C8 | C12 | C12 | C24 | C24 | D5 | D10 | C3×D5 | M5(2) | C4×D5 | C6×D5 | C8×D5 | C3×M5(2) | D5×C12 | C80⋊C2 | D5×C24 | C3×C80⋊C2 |
| kernel | C3×C80⋊C2 | C3×C5⋊2C16 | C240 | D5×C24 | C80⋊C2 | C3×C5⋊2C8 | D5×C12 | C5⋊2C16 | C80 | C8×D5 | C3×Dic5 | C6×D5 | C5⋊2C8 | C4×D5 | Dic5 | D10 | C48 | C24 | C16 | C15 | C12 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C3×C80⋊C2 ►in GL2(𝔽241) generated by
| 15 | 0 |
| 0 | 15 |
| 189 | 188 |
| 1 | 136 |
| 189 | 53 |
| 190 | 52 |
G:=sub<GL(2,GF(241))| [15,0,0,15],[189,1,188,136],[189,190,53,52] >;
C3×C80⋊C2 in GAP, Magma, Sage, TeX
C_3\times C_{80}\rtimes C_2 % in TeX
G:=Group("C3xC80:C2"); // GroupNames label
G:=SmallGroup(480,76);
// by ID
G=gap.SmallGroup(480,76);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,701,92,80,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^3=b^80=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
// generators/relations
Export