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G = C20.60D12order 480 = 25·3·5

14th non-split extension by C20 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.60D12, C60.109D4, D12.30D10, C60.127C23, Dic6.32D10, D60.48C22, Dic30.51C22, C4○D121D5, C55(C4○D24), C1511(C4○D8), C5⋊D2415C2, (C2×C10).5D12, (C2×C30).46D4, C30.78(C2×D4), (C2×C20).89D6, C52C8.35D6, C5⋊Dic1215C2, C10.48(C2×D12), Dic6⋊D515C2, D12.D515C2, (C2×C12).318D10, C31(D4.8D10), C4.32(C5⋊D12), C12.68(C5⋊D4), (C2×C60).49C22, C20.89(C22×S3), D6011C210C2, (C5×D12).35C22, C12.150(C22×D5), C22.1(C5⋊D12), (C5×Dic6).37C22, (C6×C52C8)⋊3C2, C4.75(C2×S3×D5), (C2×C52C8)⋊6S3, C6.2(C2×C5⋊D4), (C5×C4○D12)⋊4C2, C2.6(C2×C5⋊D12), (C2×C4).146(S3×D5), (C2×C6).31(C5⋊D4), (C3×C52C8).39C22, SmallGroup(480,379)

Series: Derived Chief Lower central Upper central

C1C60 — C20.60D12
C1C5C15C30C60C3×C52C8C5⋊D24 — C20.60D12
C15C30C60 — C20.60D12
C1C4C2×C4

Generators and relations for C20.60D12
 G = < a,b,c | a20=c2=1, b12=a10, bab-1=cac=a9, cbc=a10b11 >

Subgroups: 668 in 124 conjugacy classes, 44 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C5, S3 [×2], C6, C6, C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], D5, C10, C10 [×2], Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, C20 [×2], C20, D10, C2×C10, C2×C10, C24 [×2], Dic6, Dic6, C4×S3 [×2], D12, D12, C3⋊D4 [×2], C2×C12, C5×S3, D15, C30, C30, C4○D8, C52C8 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×D4 [×2], C5×Q8, C24⋊C2 [×2], D24, Dic12, C2×C24, C4○D12, C4○D12, C5×Dic3, Dic15, C60 [×2], S3×C10, D30, C2×C30, C2×C52C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4○D20, C5×C4○D4, C4○D24, C3×C52C8 [×2], C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, Dic30, C4×D15, D60, C157D4, C2×C60, D4.8D10, C5⋊D24, D12.D5, Dic6⋊D5, C5⋊Dic12, C6×C52C8, C5×C4○D12, D6011C2, C20.60D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, C4○D8, C5⋊D4 [×2], C22×D5, C2×D12, S3×D5, C2×C5⋊D4, C4○D24, C5⋊D12 [×2], C2×S3×D5, D4.8D10, C2×C5⋊D12, C20.60D12

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

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

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

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

66 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B6A6B6C8A8B8C8D10A10B10C10D10E10F10G10H12A12B12C12D15A15B20A20B20C20D20E20F20G20H20I20J24A···24H30A···30F60A···60H
order1222234444455666888810101010101010101212121215152020202020202020202024···2430···3060···60
size11212602112126022222101010102244121212122222442222441212121210···104···44···4

66 irreducible representations

dim11111111222222222222222444444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D10D10D10D12D12C4○D8C5⋊D4C5⋊D4C4○D24S3×D5C5⋊D12C2×S3×D5C5⋊D12D4.8D10C20.60D12
kernelC20.60D12C5⋊D24D12.D5Dic6⋊D5C5⋊Dic12C6×C52C8C5×C4○D12D6011C2C2×C52C8C60C2×C30C4○D12C52C8C2×C20Dic6D12C2×C12C20C2×C10C15C12C2×C6C5C2×C4C4C4C22C3C1
# reps11111111111221222224448222248

Matrix representation of C20.60D12 in GL4(𝔽241) generated by

177000
017700
0011
005051
,
113000
03200
0014163
0044100
,
012000
239000
0017695
0019465
G:=sub<GL(4,GF(241))| [177,0,0,0,0,177,0,0,0,0,1,50,0,0,1,51],[113,0,0,0,0,32,0,0,0,0,141,44,0,0,63,100],[0,239,0,0,120,0,0,0,0,0,176,194,0,0,95,65] >;

C20.60D12 in GAP, Magma, Sage, TeX

C_{20}._{60}D_{12}
% in TeX

G:=Group("C20.60D12");
// GroupNames label

G:=SmallGroup(480,379);
// by ID

G=gap.SmallGroup(480,379);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,100,346,80,1356,18822]);
// Polycyclic

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

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