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G = Dic60⋊C2order 480 = 25·3·5

14th semidirect product of Dic60 and C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.25D6, C24.1D10, Dic6014C2, D10.14D12, D12.20D10, C120.25C22, C60.121C23, Dic6.19D10, Dic5.16D12, Dic30.33C22, C6.6(D4×D5), C8.1(S3×D5), C8⋊D52S3, C24⋊C22D5, (C4×D5).3D6, (C6×D5).3D4, C52C8.1D6, (D5×Dic6)⋊9C2, C30.14(C2×D4), C10.6(C2×D12), C2.11(D5×D12), C51(C8.D6), C5⋊Dic1210C2, C31(SD16⋊D5), C151(C8.C22), (C3×Dic5).3D4, D125D5.2C2, D12.D511C2, C20.73(C22×S3), (D5×C12).26C22, (C5×D12).22C22, C12.144(C22×D5), (C5×Dic6).23C22, C4.69(C2×S3×D5), (C5×C24⋊C2)⋊2C2, (C3×C8⋊D5)⋊2C2, (C3×C52C8).19C22, SmallGroup(480,336)

Series: Derived Chief Lower central Upper central

C1C60 — Dic60⋊C2
C1C5C15C30C60D5×C12D5×Dic6 — Dic60⋊C2
C15C30C60 — Dic60⋊C2
C1C2C4C8

Generators and relations for Dic60⋊C2
 G = < a,b,c | a120=c2=1, b2=a60, bab-1=a-1, cac=a11, cbc=a30b >

Subgroups: 668 in 120 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22 [×2], C5, S3, C6, C6, C8, C8, C2×C4 [×3], D4 [×2], Q8 [×4], D5, C10, C10, Dic3 [×3], C12, C12, D6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5, Dic5 [×2], C20, C20, D10, C2×C10, C24, C24, Dic6, Dic6 [×3], C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C5×S3, C3×D5, C30, C8.C22, C52C8, C40, Dic10 [×3], C4×D5, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C24⋊C2, C24⋊C2, Dic12 [×2], C3×M4(2), C2×Dic6, C4○D12, C5×Dic3, C3×Dic5, Dic15 [×2], C60, C6×D5, S3×C10, C8⋊D5, Dic20, D4.D5, C5⋊Q16, C5×SD16, D42D5, Q8×D5, C8.D6, C3×C52C8, C120, D5×Dic3, S3×Dic5, C15⋊D4, C15⋊Q8, D5×C12, C5×Dic6, C5×D12, Dic30 [×2], SD16⋊D5, D12.D5, C5⋊Dic12, C3×C8⋊D5, C5×C24⋊C2, Dic60, D5×Dic6, D125D5, Dic60⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, C8.C22, C22×D5, C2×D12, S3×D5, D4×D5, C8.D6, C2×S3×D5, SD16⋊D5, D5×D12, Dic60⋊C2

Smallest permutation representation of Dic60⋊C2
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)
(1 192 61 132)(2 191 62 131)(3 190 63 130)(4 189 64 129)(5 188 65 128)(6 187 66 127)(7 186 67 126)(8 185 68 125)(9 184 69 124)(10 183 70 123)(11 182 71 122)(12 181 72 121)(13 180 73 240)(14 179 74 239)(15 178 75 238)(16 177 76 237)(17 176 77 236)(18 175 78 235)(19 174 79 234)(20 173 80 233)(21 172 81 232)(22 171 82 231)(23 170 83 230)(24 169 84 229)(25 168 85 228)(26 167 86 227)(27 166 87 226)(28 165 88 225)(29 164 89 224)(30 163 90 223)(31 162 91 222)(32 161 92 221)(33 160 93 220)(34 159 94 219)(35 158 95 218)(36 157 96 217)(37 156 97 216)(38 155 98 215)(39 154 99 214)(40 153 100 213)(41 152 101 212)(42 151 102 211)(43 150 103 210)(44 149 104 209)(45 148 105 208)(46 147 106 207)(47 146 107 206)(48 145 108 205)(49 144 109 204)(50 143 110 203)(51 142 111 202)(52 141 112 201)(53 140 113 200)(54 139 114 199)(55 138 115 198)(56 137 116 197)(57 136 117 196)(58 135 118 195)(59 134 119 194)(60 133 120 193)
(1 177)(2 188)(3 199)(4 210)(5 221)(6 232)(7 123)(8 134)(9 145)(10 156)(11 167)(12 178)(13 189)(14 200)(15 211)(16 222)(17 233)(18 124)(19 135)(20 146)(21 157)(22 168)(23 179)(24 190)(25 201)(26 212)(27 223)(28 234)(29 125)(30 136)(31 147)(32 158)(33 169)(34 180)(35 191)(36 202)(37 213)(38 224)(39 235)(40 126)(41 137)(42 148)(43 159)(44 170)(45 181)(46 192)(47 203)(48 214)(49 225)(50 236)(51 127)(52 138)(53 149)(54 160)(55 171)(56 182)(57 193)(58 204)(59 215)(60 226)(61 237)(62 128)(63 139)(64 150)(65 161)(66 172)(67 183)(68 194)(69 205)(70 216)(71 227)(72 238)(73 129)(74 140)(75 151)(76 162)(77 173)(78 184)(79 195)(80 206)(81 217)(82 228)(83 239)(84 130)(85 141)(86 152)(87 163)(88 174)(89 185)(90 196)(91 207)(92 218)(93 229)(94 240)(95 131)(96 142)(97 153)(98 164)(99 175)(100 186)(101 197)(102 208)(103 219)(104 230)(105 121)(106 132)(107 143)(108 154)(109 165)(110 176)(111 187)(112 198)(113 209)(114 220)(115 231)(116 122)(117 133)(118 144)(119 155)(120 166)

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

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

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

51 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E5A5B6A6B8A8B10A10B10C10D12A12B12C15A15B20A20B20C20D24A24B24C24D30A30B40A40B40C40D60A60B60C60D120A···120H
order1222344444556688101010101212121515202020202424242430304040404060606060120···120
size11101222101260602222042022242422204444242444202044444444444···4

51 irreducible representations

dim1111111122222222222244444444
type++++++++++++++++++++-++-+-+-
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10D12D12C8.C22S3×D5D4×D5C8.D6C2×S3×D5SD16⋊D5D5×D12Dic60⋊C2
kernelDic60⋊C2D12.D5C5⋊Dic12C3×C8⋊D5C5×C24⋊C2Dic60D5×Dic6D125D5C8⋊D5C3×Dic5C6×D5C24⋊C2C52C8C40C4×D5C24Dic6D12Dic5D10C15C8C6C5C4C3C2C1
# reps1111111111121112222212222448

Matrix representation of Dic60⋊C2 in GL6(𝔽241)

56440000
11500000
00183179183179
0062336233
00293100
0021010400
,
1731120000
232680000
0022101510
00762010190
002160200
009525165221
,
2202060000
47210000
001392002204
004110237239
0010314310241
0098138200139

G:=sub<GL(6,GF(241))| [56,115,0,0,0,0,44,0,0,0,0,0,0,0,183,62,29,210,0,0,179,33,31,104,0,0,183,62,0,0,0,0,179,33,0,0],[173,232,0,0,0,0,112,68,0,0,0,0,0,0,221,76,216,95,0,0,0,20,0,25,0,0,151,101,20,165,0,0,0,90,0,221],[220,47,0,0,0,0,206,21,0,0,0,0,0,0,139,41,103,98,0,0,200,102,143,138,0,0,2,37,102,200,0,0,204,239,41,139] >;

Dic60⋊C2 in GAP, Magma, Sage, TeX

{\rm Dic}_{60}\rtimes C_2
% in TeX

G:=Group("Dic60:C2");
// GroupNames label

G:=SmallGroup(480,336);
// by ID

G=gap.SmallGroup(480,336);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,422,135,58,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^120=c^2=1,b^2=a^60,b*a*b^-1=a^-1,c*a*c=a^11,c*b*c=a^30*b>;
// generators/relations

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