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G = C20.D12order 480 = 25·3·5

18th non-split extension by C20 of D12 acting via D12/C6=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.87D4, C20.18D12, C60.133C23, Dic6.34D10, D60.51C22, Dic30.53C22, C52C8.3D6, (C2×Dic6)⋊9D5, (C2×C30).64D4, C30.96(C2×D4), C54(C8.D6), C4.Dic59S3, (C10×Dic6)⋊2C2, C5⋊Dic1213C2, C10.54(C2×D12), (C2×C20).103D6, (C2×C10).43D12, Dic6⋊D514C2, (C2×C12).104D10, C31(C20.C23), C4.24(C5⋊D12), C12.33(C5⋊D4), C1511(C8.C22), (C2×C60).27C22, D6011C2.2C2, C20.100(C22×S3), C12.156(C22×D5), C22.5(C5⋊D12), (C5×Dic6).39C22, C4.81(C2×S3×D5), C6.8(C2×C5⋊D4), (C2×C4).17(S3×D5), C2.12(C2×C5⋊D12), (C3×C4.Dic5)⋊3C2, (C2×C6).14(C5⋊D4), (C3×C52C8).21C22, SmallGroup(480,397)

Series: Derived Chief Lower central Upper central

C1C60 — C20.D12
C1C5C15C30C60C3×C52C8C5⋊Dic12 — C20.D12
C15C30C60 — C20.D12
C1C2C2×C4

Generators and relations for C20.D12
 G = < a,b,c | a20=1, b12=c2=a10, bab-1=a-1, cac-1=a9, cbc-1=a10b11 >

Subgroups: 636 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 [×3], C12 [×2], D6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5, C20 [×2], C20 [×2], D10, C2×C10, C24 [×2], Dic6 [×2], Dic6 [×2], C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, D15, C30, C30, C8.C22, C52C8 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×Q8 [×3], C24⋊C2 [×2], Dic12 [×2], C3×M4(2), C2×Dic6, C4○D12, C5×Dic3 [×2], Dic15, C60 [×2], D30, C2×C30, C4.Dic5, Q8⋊D5 [×2], C5⋊Q16 [×2], C4○D20, Q8×C10, C8.D6, C3×C52C8 [×2], C5×Dic6 [×2], C5×Dic6, C10×Dic3, Dic30, C4×D15, D60, C157D4, C2×C60, C20.C23, Dic6⋊D5 [×2], C5⋊Dic12 [×2], C3×C4.Dic5, C10×Dic6, D6011C2, C20.D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, C8.C22, C5⋊D4 [×2], C22×D5, C2×D12, S3×D5, C2×C5⋊D4, C8.D6, C5⋊D12 [×2], C2×S3×D5, C20.C23, C2×C5⋊D12, C20.D12

Smallest permutation representation of C20.D12
On 240 points
Generators in S240
(1 230 86 39 49 19 224 80 33 67 13 218 74 27 61 7 236 92 45 55)(2 56 46 93 237 8 62 28 75 219 14 68 34 81 225 20 50 40 87 231)(3 232 88 41 51 21 226 82 35 69 15 220 76 29 63 9 238 94 47 57)(4 58 48 95 239 10 64 30 77 221 16 70 36 83 227 22 52 42 89 233)(5 234 90 43 53 23 228 84 37 71 17 222 78 31 65 11 240 96 25 59)(6 60 26 73 217 12 66 32 79 223 18 72 38 85 229 24 54 44 91 235)(97 128 165 174 213 103 134 147 180 195 109 140 153 186 201 115 122 159 192 207)(98 208 169 160 123 116 202 187 154 141 110 196 181 148 135 104 214 175 166 129)(99 130 167 176 215 105 136 149 182 197 111 142 155 188 203 117 124 161 170 209)(100 210 171 162 125 118 204 189 156 143 112 198 183 150 137 106 216 177 168 131)(101 132 145 178 193 107 138 151 184 199 113 144 157 190 205 119 126 163 172 211)(102 212 173 164 127 120 206 191 158 121 114 200 185 152 139 108 194 179 146 133)
(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)
(1 167 13 155)(2 166 14 154)(3 165 15 153)(4 164 16 152)(5 163 17 151)(6 162 18 150)(7 161 19 149)(8 160 20 148)(9 159 21 147)(10 158 22 146)(11 157 23 145)(12 156 24 168)(25 211 37 199)(26 210 38 198)(27 209 39 197)(28 208 40 196)(29 207 41 195)(30 206 42 194)(31 205 43 193)(32 204 44 216)(33 203 45 215)(34 202 46 214)(35 201 47 213)(36 200 48 212)(49 170 61 182)(50 169 62 181)(51 192 63 180)(52 191 64 179)(53 190 65 178)(54 189 66 177)(55 188 67 176)(56 187 68 175)(57 186 69 174)(58 185 70 173)(59 184 71 172)(60 183 72 171)(73 112 85 100)(74 111 86 99)(75 110 87 98)(76 109 88 97)(77 108 89 120)(78 107 90 119)(79 106 91 118)(80 105 92 117)(81 104 93 116)(82 103 94 115)(83 102 95 114)(84 101 96 113)(121 239 133 227)(122 238 134 226)(123 237 135 225)(124 236 136 224)(125 235 137 223)(126 234 138 222)(127 233 139 221)(128 232 140 220)(129 231 141 219)(130 230 142 218)(131 229 143 217)(132 228 144 240)

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

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

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

57 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E5A5B6A6B8A8B10A···10F12A12B12C15A15B20A20B20C20D20E···20L24A24B24C24D30A···30F60A···60H
order122234444455668810···1012121215152020202020···202424242430···3060···60
size11260222121260222420202···222444444412···12202020204···44···4

57 irreducible representations

dim11111122222222222244444444
type++++++++++++++++-+-+++
imageC1C2C2C2C2C2S3D4D4D5D6D6D10D10D12D12C5⋊D4C5⋊D4C8.C22S3×D5C8.D6C5⋊D12C2×S3×D5C5⋊D12C20.C23C20.D12
kernelC20.D12Dic6⋊D5C5⋊Dic12C3×C4.Dic5C10×Dic6D6011C2C4.Dic5C60C2×C30C2×Dic6C52C8C2×C20Dic6C2×C12C20C2×C10C12C2×C6C15C2×C4C5C4C4C22C3C1
# reps12211111122142224412222248

Matrix representation of C20.D12 in GL8(𝔽241)

190240000000
191240000000
24001892400000
511100000
0000367200
000020520500
00000087174
000000154154
,
93107152270000
1161341852120000
142215401570000
41752312150000
00000010
00000001
000024023900
00001100
,
14730000000
6794000000
13042211630000
150102177300000
000000962
00000022232
000096200
00002223200

G:=sub<GL(8,GF(241))| [190,191,240,51,0,0,0,0,240,240,0,1,0,0,0,0,0,0,189,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,36,205,0,0,0,0,0,0,72,205,0,0,0,0,0,0,0,0,87,154,0,0,0,0,0,0,174,154],[93,116,142,4,0,0,0,0,107,134,215,175,0,0,0,0,15,185,40,231,0,0,0,0,227,212,157,215,0,0,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,0,239,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[147,67,130,150,0,0,0,0,30,94,42,102,0,0,0,0,0,0,211,177,0,0,0,0,0,0,63,30,0,0,0,0,0,0,0,0,0,0,9,22,0,0,0,0,0,0,62,232,0,0,0,0,9,22,0,0,0,0,0,0,62,232,0,0] >;

C20.D12 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(480,397);
// by ID

G=gap.SmallGroup(480,397);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,120,422,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^20=1,b^12=c^2=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^10*b^11>;
// generators/relations

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