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G = C12.D20order 480 = 25·3·5

17th non-split extension by C12 of D20 acting via D20/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.85D4, C12.17D20, C60.108C23, Dic10.34D6, D60.46C22, Dic30.48C22, C3⋊C8.4D10, (C2×C20).98D6, (C2×C6).42D20, C30.90(C2×D4), (C2×C30).58D4, C6.53(C2×D20), (C2×Dic10)⋊9S3, (C6×Dic10)⋊2C2, C3⋊Dic2014C2, C34(C8.D10), C4.Dic310D5, (C2×C12).101D10, C15⋊SD1614C2, C4.24(C3⋊D20), C20.31(C3⋊D4), C51(Q8.11D6), C1510(C8.C22), (C2×C60).34C22, C12.99(C22×D5), D6011C2.3C2, C20.158(C22×S3), C22.5(C3⋊D20), (C3×Dic10).39C22, C4.107(C2×S3×D5), (C2×C4).16(S3×D5), C10.8(C2×C3⋊D4), C2.12(C2×C3⋊D20), (C5×C3⋊C8).22C22, (C5×C4.Dic3)⋊3C2, (C2×C10).14(C3⋊D4), SmallGroup(480,391)

Series: Derived Chief Lower central Upper central

C1C60 — C12.D20
C1C5C15C30C60C3×Dic10C15⋊SD16 — C12.D20
C15C30C60 — C12.D20
C1C2C2×C4

Generators and relations for C12.D20
 G = < a,b,c | a12=c2=1, b20=a6, bab-1=a-1, cac=a5, cbc=b19 >

Subgroups: 668 in 120 conjugacy classes, 44 normal (32 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, S3, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], D5, C10, C10, Dic3, C12 [×2], C12 [×2], D6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5 [×3], C20 [×2], D10, C2×C10, C3⋊C8 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×Q8 [×3], D15, C30, C30, C8.C22, C40 [×2], Dic10 [×2], Dic10 [×2], C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C4.Dic3, Q82S3 [×2], C3⋊Q16 [×2], C4○D12, C6×Q8, C3×Dic5 [×2], Dic15, C60 [×2], D30, C2×C30, C40⋊C2 [×2], Dic20 [×2], C5×M4(2), C2×Dic10, C4○D20, Q8.11D6, C5×C3⋊C8 [×2], C3×Dic10 [×2], C3×Dic10, C6×Dic5, Dic30, C4×D15, D60, C157D4, C2×C60, C8.D10, C15⋊SD16 [×2], C3⋊Dic20 [×2], C5×C4.Dic3, C6×Dic10, D6011C2, C12.D20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C8.C22, D20 [×2], C22×D5, C2×C3⋊D4, S3×D5, C2×D20, Q8.11D6, C3⋊D20 [×2], C2×S3×D5, C8.D10, C2×C3⋊D20, C12.D20

Smallest permutation representation of C12.D20
On 240 points
Generators in S240
(1 236 120 31 226 110 21 216 100 11 206 90)(2 91 207 12 101 217 22 111 227 32 81 237)(3 238 82 33 228 112 23 218 102 13 208 92)(4 93 209 14 103 219 24 113 229 34 83 239)(5 240 84 35 230 114 25 220 104 15 210 94)(6 95 211 16 105 221 26 115 231 36 85 201)(7 202 86 37 232 116 27 222 106 17 212 96)(8 97 213 18 107 223 28 117 233 38 87 203)(9 204 88 39 234 118 29 224 108 19 214 98)(10 99 215 20 109 225 30 119 235 40 89 205)(41 159 177 71 149 167 61 139 197 51 129 187)(42 188 130 52 198 140 62 168 150 72 178 160)(43 121 179 73 151 169 63 141 199 53 131 189)(44 190 132 54 200 142 64 170 152 74 180 122)(45 123 181 75 153 171 65 143 161 55 133 191)(46 192 134 56 162 144 66 172 154 76 182 124)(47 125 183 77 155 173 67 145 163 57 135 193)(48 194 136 58 164 146 68 174 156 78 184 126)(49 127 185 79 157 175 69 147 165 59 137 195)(50 196 138 60 166 148 70 176 158 80 186 128)
(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)
(1 158)(2 137)(3 156)(4 135)(5 154)(6 133)(7 152)(8 131)(9 150)(10 129)(11 148)(12 127)(13 146)(14 125)(15 144)(16 123)(17 142)(18 121)(19 140)(20 159)(21 138)(22 157)(23 136)(24 155)(25 134)(26 153)(27 132)(28 151)(29 130)(30 149)(31 128)(32 147)(33 126)(34 145)(35 124)(36 143)(37 122)(38 141)(39 160)(40 139)(41 89)(42 108)(43 87)(44 106)(45 85)(46 104)(47 83)(48 102)(49 81)(50 100)(51 119)(52 98)(53 117)(54 96)(55 115)(56 94)(57 113)(58 92)(59 111)(60 90)(61 109)(62 88)(63 107)(64 86)(65 105)(66 84)(67 103)(68 82)(69 101)(70 120)(71 99)(72 118)(73 97)(74 116)(75 95)(76 114)(77 93)(78 112)(79 91)(80 110)(161 211)(162 230)(163 209)(164 228)(165 207)(166 226)(167 205)(168 224)(169 203)(170 222)(171 201)(172 220)(173 239)(174 218)(175 237)(176 216)(177 235)(178 214)(179 233)(180 212)(181 231)(182 210)(183 229)(184 208)(185 227)(186 206)(187 225)(188 204)(189 223)(190 202)(191 221)(192 240)(193 219)(194 238)(195 217)(196 236)(197 215)(198 234)(199 213)(200 232)

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

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

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

57 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E5A5B6A6B6C8A8B10A10B10C10D12A12B12C12D12E12F15A15B20A20B20C20D20E20F30A···30F40A···40H60A···60H
order1222344444556668810101010121212121212151520202020202030···3040···4060···60
size1126022220206022222121222444420202020442222444···412···124···4

57 irreducible representations

dim11111122222222222244444444
type++++++++++++++++-++++-
imageC1C2C2C2C2C2S3D4D4D5D6D6D10D10C3⋊D4C3⋊D4D20D20C8.C22S3×D5Q8.11D6C3⋊D20C2×S3×D5C3⋊D20C8.D10C12.D20
kernelC12.D20C15⋊SD16C3⋊Dic20C5×C4.Dic3C6×Dic10D6011C2C2×Dic10C60C2×C30C4.Dic3Dic10C2×C20C3⋊C8C2×C12C20C2×C10C12C2×C6C15C2×C4C5C4C4C22C3C1
# reps12211111122142224412222248

Matrix representation of C12.D20 in GL6(𝔽241)

2402400000
100000
0044300
0023819700
0000197238
0000344
,
20420000
39370000
00000239
00002139
001222200
002193900
,
20420000
39370000
000023577
0000126
001224100
0023811900

G:=sub<GL(6,GF(241))| [240,1,0,0,0,0,240,0,0,0,0,0,0,0,44,238,0,0,0,0,3,197,0,0,0,0,0,0,197,3,0,0,0,0,238,44],[204,39,0,0,0,0,2,37,0,0,0,0,0,0,0,0,122,219,0,0,0,0,22,39,0,0,0,2,0,0,0,0,239,139,0,0],[204,39,0,0,0,0,2,37,0,0,0,0,0,0,0,0,122,238,0,0,0,0,41,119,0,0,235,12,0,0,0,0,77,6,0,0] >;

C12.D20 in GAP, Magma, Sage, TeX

C_{12}.D_{20}
% in TeX

G:=Group("C12.D20");
// GroupNames label

G:=SmallGroup(480,391);
// by ID

G=gap.SmallGroup(480,391);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,64,219,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^12=c^2=1,b^20=a^6,b*a*b^-1=a^-1,c*a*c=a^5,c*b*c=b^19>;
// generators/relations

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