Copied to
clipboard

G = C2×D120order 480 = 25·3·5

Direct product of C2 and D120

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D120, C304D8, C61D40, C87D30, C101D24, C4029D6, C4.7D60, C2429D10, C60.163D4, C12.32D20, C20.32D12, C12036C22, D6020C22, C22.13D60, C60.245C23, C32(C2×D40), C52(C2×D24), (C2×C40)⋊5S3, (C2×C24)⋊5D5, (C2×C8)⋊3D15, C1510(C2×D8), (C2×C120)⋊9C2, (C2×D60)⋊8C2, C2.12(C2×D60), C6.38(C2×D20), (C2×C4).81D30, (C2×C6).19D20, C10.39(C2×D12), (C2×C20).391D6, (C2×C10).19D12, (C2×C30).104D4, C30.266(C2×D4), (C2×C12).397D10, C4.26(C22×D15), C20.216(C22×S3), (C2×C60).478C22, C12.218(C22×D5), SmallGroup(480,868)

Series: Derived Chief Lower central Upper central

C1C60 — C2×D120
C1C5C15C30C60D60C2×D60 — C2×D120
C15C30C60 — C2×D120
C1C22C2×C4C2×C8

Generators and relations for C2×D120
 G = < a,b,c | a2=b120=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 1460 in 152 conjugacy classes, 55 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], C5, S3 [×4], C6, C6 [×2], C8 [×2], C2×C4, D4 [×6], C23 [×2], D5 [×4], C10, C10 [×2], C12 [×2], D6 [×8], C2×C6, C15, C2×C8, D8 [×4], C2×D4 [×2], C20 [×2], D10 [×8], C2×C10, C24 [×2], D12 [×6], C2×C12, C22×S3 [×2], D15 [×4], C30, C30 [×2], C2×D8, C40 [×2], D20 [×6], C2×C20, C22×D5 [×2], D24 [×4], C2×C24, C2×D12 [×2], C60 [×2], D30 [×8], C2×C30, D40 [×4], C2×C40, C2×D20 [×2], C2×D24, C120 [×2], D60 [×4], D60 [×2], C2×C60, C22×D15 [×2], C2×D40, D120 [×4], C2×C120, C2×D60 [×2], C2×D120
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], D8 [×2], C2×D4, D10 [×3], D12 [×2], C22×S3, D15, C2×D8, D20 [×2], C22×D5, D24 [×2], C2×D12, D30 [×3], D40 [×2], C2×D20, C2×D24, D60 [×2], C22×D15, C2×D40, D120 [×2], C2×D60, C2×D120

Smallest permutation representation of C2×D120
On 240 points
Generators in S240
(1 130)(2 131)(3 132)(4 133)(5 134)(6 135)(7 136)(8 137)(9 138)(10 139)(11 140)(12 141)(13 142)(14 143)(15 144)(16 145)(17 146)(18 147)(19 148)(20 149)(21 150)(22 151)(23 152)(24 153)(25 154)(26 155)(27 156)(28 157)(29 158)(30 159)(31 160)(32 161)(33 162)(34 163)(35 164)(36 165)(37 166)(38 167)(39 168)(40 169)(41 170)(42 171)(43 172)(44 173)(45 174)(46 175)(47 176)(48 177)(49 178)(50 179)(51 180)(52 181)(53 182)(54 183)(55 184)(56 185)(57 186)(58 187)(59 188)(60 189)(61 190)(62 191)(63 192)(64 193)(65 194)(66 195)(67 196)(68 197)(69 198)(70 199)(71 200)(72 201)(73 202)(74 203)(75 204)(76 205)(77 206)(78 207)(79 208)(80 209)(81 210)(82 211)(83 212)(84 213)(85 214)(86 215)(87 216)(88 217)(89 218)(90 219)(91 220)(92 221)(93 222)(94 223)(95 224)(96 225)(97 226)(98 227)(99 228)(100 229)(101 230)(102 231)(103 232)(104 233)(105 234)(106 235)(107 236)(108 237)(109 238)(110 239)(111 240)(112 121)(113 122)(114 123)(115 124)(116 125)(117 126)(118 127)(119 128)(120 129)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 120)(17 119)(18 118)(19 117)(20 116)(21 115)(22 114)(23 113)(24 112)(25 111)(26 110)(27 109)(28 108)(29 107)(30 106)(31 105)(32 104)(33 103)(34 102)(35 101)(36 100)(37 99)(38 98)(39 97)(40 96)(41 95)(42 94)(43 93)(44 92)(45 91)(46 90)(47 89)(48 88)(49 87)(50 86)(51 85)(52 84)(53 83)(54 82)(55 81)(56 80)(57 79)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(121 153)(122 152)(123 151)(124 150)(125 149)(126 148)(127 147)(128 146)(129 145)(130 144)(131 143)(132 142)(133 141)(134 140)(135 139)(136 138)(154 240)(155 239)(156 238)(157 237)(158 236)(159 235)(160 234)(161 233)(162 232)(163 231)(164 230)(165 229)(166 228)(167 227)(168 226)(169 225)(170 224)(171 223)(172 222)(173 221)(174 220)(175 219)(176 218)(177 217)(178 216)(179 215)(180 214)(181 213)(182 212)(183 211)(184 210)(185 209)(186 208)(187 207)(188 206)(189 205)(190 204)(191 203)(192 202)(193 201)(194 200)(195 199)(196 198)

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

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

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

126 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B5A5B6A6B6C8A8B8C8D10A···10F12A12B12C12D15A15B15C15D20A···20H24A···24H30A···30L40A···40P60A···60P120A···120AF
order1222222234455666888810···10121212121515151520···2024···2430···3040···4060···60120···120
size1111606060602222222222222···2222222222···22···22···22···22···22···2

126 irreducible representations

dim1111222222222222222222222
type+++++++++++++++++++++++++
imageC1C2C2C2S3D4D4D5D6D6D8D10D10D12D12D15D20D20D24D30D30D40D60D60D120
kernelC2×D120D120C2×C120C2×D60C2×C40C60C2×C30C2×C24C40C2×C20C30C24C2×C12C20C2×C10C2×C8C12C2×C6C10C8C2×C4C6C4C22C2
# reps141211122144222444884168832

Matrix representation of C2×D120 in GL4(𝔽241) generated by

240000
024000
0010
0001
,
124000
1000
0039138
00157123
,
240000
240100
00113122
00164128
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[1,1,0,0,240,0,0,0,0,0,39,157,0,0,138,123],[240,240,0,0,0,1,0,0,0,0,113,164,0,0,122,128] >;

C2×D120 in GAP, Magma, Sage, TeX

C_2\times D_{120}
% in TeX

G:=Group("C2xD120");
// GroupNames label

G:=SmallGroup(480,868);
// by ID

G=gap.SmallGroup(480,868);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,142,675,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^2=b^120=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

׿
×
𝔽