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G = C202D12order 480 = 25·3·5

2nd semidirect product of C20 and D12 acting via D12/C6=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C202D12, C6012D4, D3014D4, C6.50(D4×D5), (C2×D12)⋊11D5, C55(C12⋊D4), C124(C5⋊D4), C43(C5⋊D12), C31(C202D4), C4⋊Dic515S3, C10.51(S3×D4), (C10×D12)⋊11C2, C1515(C4⋊D4), D6⋊Dic521C2, C30.159(C2×D4), (C2×C20).134D6, C10.63(C2×D12), C30.96(C4○D4), (C2×C12).135D10, (C2×Dic5).49D6, C6.15(D42D5), C2.26(C20⋊D6), (C2×C30).156C23, (C2×C60).205C22, (C22×S3).23D10, C2.19(D12⋊D5), C10.37(Q83S3), (C6×Dic5).94C22, (C2×Dic15).217C22, (C22×D15).110C22, (C2×C4×D15)⋊23C2, (C2×C5⋊D12)⋊8C2, C6.18(C2×C5⋊D4), (C3×C4⋊Dic5)⋊12C2, (C2×C4).215(S3×D5), C2.21(C2×C5⋊D12), C22.208(C2×S3×D5), (S3×C2×C10).39C22, (C2×C6).168(C22×D5), (C2×C10).168(C22×S3), SmallGroup(480,542)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C202D12
C1C5C15C30C2×C30C6×Dic5C2×C5⋊D12 — C202D12
C15C2×C30 — C202D12
C1C22C2×C4

Generators and relations for C202D12
 G = < a,b,c | a60=b4=c2=1, bab-1=a19, cac=a29, cbc=b-1 >

Subgroups: 1100 in 188 conjugacy classes, 54 normal (32 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×10], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], D5 [×2], C10 [×3], C10 [×2], Dic3, C12 [×2], C12 [×2], D6 [×10], C2×C6, C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5 [×3], C20 [×2], D10 [×4], C2×C10, C2×C10 [×6], C4×S3 [×2], D12 [×6], C2×Dic3, C2×C12, C2×C12 [×2], C22×S3 [×2], C22×S3, C5×S3 [×2], D15 [×2], C30 [×3], C4⋊D4, C4×D5 [×2], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×C10 [×2], D6⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C2×D12, C2×D12 [×2], C3×Dic5 [×2], Dic15, C60 [×2], S3×C10 [×6], D30 [×2], D30 [×2], C2×C30, C4⋊Dic5, C23.D5 [×2], C2×C4×D5, C2×C5⋊D4 [×2], D4×C10, C12⋊D4, C5⋊D12 [×4], C6×Dic5 [×2], C5×D12 [×2], C4×D15 [×2], C2×Dic15, C2×C60, S3×C2×C10 [×2], C22×D15, C202D4, D6⋊Dic5 [×2], C3×C4⋊Dic5, C2×C5⋊D12 [×2], C10×D12, C2×C4×D15, C202D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], D12 [×2], C22×S3, C4⋊D4, C5⋊D4 [×2], C22×D5, C2×D12, S3×D4, Q83S3, S3×D5, D4×D5, D42D5, C2×C5⋊D4, C12⋊D4, C5⋊D12 [×2], C2×S3×D5, C202D4, D12⋊D5, C20⋊D6, C2×C5⋊D12, C202D12

Smallest permutation representation of C202D12
On 240 points
Generators in S240
(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)
(1 184 87 151)(2 203 88 170)(3 222 89 129)(4 181 90 148)(5 200 91 167)(6 219 92 126)(7 238 93 145)(8 197 94 164)(9 216 95 123)(10 235 96 142)(11 194 97 161)(12 213 98 180)(13 232 99 139)(14 191 100 158)(15 210 101 177)(16 229 102 136)(17 188 103 155)(18 207 104 174)(19 226 105 133)(20 185 106 152)(21 204 107 171)(22 223 108 130)(23 182 109 149)(24 201 110 168)(25 220 111 127)(26 239 112 146)(27 198 113 165)(28 217 114 124)(29 236 115 143)(30 195 116 162)(31 214 117 121)(32 233 118 140)(33 192 119 159)(34 211 120 178)(35 230 61 137)(36 189 62 156)(37 208 63 175)(38 227 64 134)(39 186 65 153)(40 205 66 172)(41 224 67 131)(42 183 68 150)(43 202 69 169)(44 221 70 128)(45 240 71 147)(46 199 72 166)(47 218 73 125)(48 237 74 144)(49 196 75 163)(50 215 76 122)(51 234 77 141)(52 193 78 160)(53 212 79 179)(54 231 80 138)(55 190 81 157)(56 209 82 176)(57 228 83 135)(58 187 84 154)(59 206 85 173)(60 225 86 132)
(2 30)(3 59)(4 28)(5 57)(6 26)(7 55)(8 24)(9 53)(10 22)(11 51)(12 20)(13 49)(14 18)(15 47)(17 45)(19 43)(21 41)(23 39)(25 37)(27 35)(29 33)(32 60)(34 58)(36 56)(38 54)(40 52)(42 50)(44 48)(61 113)(62 82)(63 111)(64 80)(65 109)(66 78)(67 107)(68 76)(69 105)(70 74)(71 103)(73 101)(75 99)(77 97)(79 95)(81 93)(83 91)(84 120)(85 89)(86 118)(88 116)(90 114)(92 112)(94 110)(96 108)(98 106)(100 104)(115 119)(121 214)(122 183)(123 212)(124 181)(125 210)(126 239)(127 208)(128 237)(129 206)(130 235)(131 204)(132 233)(133 202)(134 231)(135 200)(136 229)(137 198)(138 227)(139 196)(140 225)(141 194)(142 223)(143 192)(144 221)(145 190)(146 219)(147 188)(148 217)(149 186)(150 215)(151 184)(152 213)(153 182)(154 211)(155 240)(156 209)(157 238)(158 207)(159 236)(160 205)(161 234)(162 203)(163 232)(164 201)(165 230)(166 199)(167 228)(168 197)(169 226)(170 195)(171 224)(172 193)(173 222)(174 191)(175 220)(176 189)(177 218)(178 187)(179 216)(180 185)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B6A6B6C10A···10F10G···10N12A12B12C12D12E12F15A15B20A20B20C20D30A···30F60A···60H
order1222222234444445566610···1010···1012121212121215152020202030···3060···60
size11111212303022220203030222222···212···1244202020204444444···44···4

60 irreducible representations

dim11111122222222222444444444
type+++++++++++++++++++-++
imageC1C2C2C2C2C2S3D4D4D5D6D6C4○D4D10D10D12C5⋊D4S3×D4Q83S3S3×D5D4×D5D42D5C5⋊D12C2×S3×D5D12⋊D5C20⋊D6
kernelC202D12D6⋊Dic5C3×C4⋊Dic5C2×C5⋊D12C10×D12C2×C4×D15C4⋊Dic5C60D30C2×D12C2×Dic5C2×C20C30C2×C12C22×S3C20C12C10C10C2×C4C6C6C4C22C2C2
# reps12121112222122448112224244

Matrix representation of C202D12 in GL6(𝔽61)

44440000
17600000
0006000
0016000
0000120
0000660
,
100000
17600000
00231500
00463800
0000120
0000060
,
100000
17600000
0006000
0060000
000010
000001

G:=sub<GL(6,GF(61))| [44,17,0,0,0,0,44,60,0,0,0,0,0,0,0,1,0,0,0,0,60,60,0,0,0,0,0,0,1,6,0,0,0,0,20,60],[1,17,0,0,0,0,0,60,0,0,0,0,0,0,23,46,0,0,0,0,15,38,0,0,0,0,0,0,1,0,0,0,0,0,20,60],[1,17,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C202D12 in GAP, Magma, Sage, TeX

C_{20}\rtimes_2D_{12}
% in TeX

G:=Group("C20:2D12");
// GroupNames label

G:=SmallGroup(480,542);
// by ID

G=gap.SmallGroup(480,542);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,422,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^19,c*a*c=a^29,c*b*c=b^-1>;
// generators/relations

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