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## G = Dic60⋊C2order 480 = 25·3·5

### 14th semidirect product of Dic60 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — Dic60⋊C2
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D5×Dic6 — Dic60⋊C2
 Lower central C15 — C30 — C60 — Dic60⋊C2
 Upper central C1 — C2 — C4 — C8

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, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, C24, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C5×S3, C3×D5, C30, C8.C22, C52C8, C40, Dic10, C4×D5, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C24⋊C2, C24⋊C2, Dic12, C3×M4(2), C2×Dic6, C4○D12, C5×Dic3, C3×Dic5, Dic15, 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, SD16⋊D5, D12.D5, C5⋊Dic12, C3×C8⋊D5, C5×C24⋊C2, Dic60, D5×Dic6, D125D5, Dic60⋊C2
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, 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)
(1 198 61 138)(2 197 62 137)(3 196 63 136)(4 195 64 135)(5 194 65 134)(6 193 66 133)(7 192 67 132)(8 191 68 131)(9 190 69 130)(10 189 70 129)(11 188 71 128)(12 187 72 127)(13 186 73 126)(14 185 74 125)(15 184 75 124)(16 183 76 123)(17 182 77 122)(18 181 78 121)(19 180 79 240)(20 179 80 239)(21 178 81 238)(22 177 82 237)(23 176 83 236)(24 175 84 235)(25 174 85 234)(26 173 86 233)(27 172 87 232)(28 171 88 231)(29 170 89 230)(30 169 90 229)(31 168 91 228)(32 167 92 227)(33 166 93 226)(34 165 94 225)(35 164 95 224)(36 163 96 223)(37 162 97 222)(38 161 98 221)(39 160 99 220)(40 159 100 219)(41 158 101 218)(42 157 102 217)(43 156 103 216)(44 155 104 215)(45 154 105 214)(46 153 106 213)(47 152 107 212)(48 151 108 211)(49 150 109 210)(50 149 110 209)(51 148 111 208)(52 147 112 207)(53 146 113 206)(54 145 114 205)(55 144 115 204)(56 143 116 203)(57 142 117 202)(58 141 118 201)(59 140 119 200)(60 139 120 199)
(1 183)(2 194)(3 205)(4 216)(5 227)(6 238)(7 129)(8 140)(9 151)(10 162)(11 173)(12 184)(13 195)(14 206)(15 217)(16 228)(17 239)(18 130)(19 141)(20 152)(21 163)(22 174)(23 185)(24 196)(25 207)(26 218)(27 229)(28 240)(29 131)(30 142)(31 153)(32 164)(33 175)(34 186)(35 197)(36 208)(37 219)(38 230)(39 121)(40 132)(41 143)(42 154)(43 165)(44 176)(45 187)(46 198)(47 209)(48 220)(49 231)(50 122)(51 133)(52 144)(53 155)(54 166)(55 177)(56 188)(57 199)(58 210)(59 221)(60 232)(61 123)(62 134)(63 145)(64 156)(65 167)(66 178)(67 189)(68 200)(69 211)(70 222)(71 233)(72 124)(73 135)(74 146)(75 157)(76 168)(77 179)(78 190)(79 201)(80 212)(81 223)(82 234)(83 125)(84 136)(85 147)(86 158)(87 169)(88 180)(89 191)(90 202)(91 213)(92 224)(93 235)(94 126)(95 137)(96 148)(97 159)(98 170)(99 181)(100 192)(101 203)(102 214)(103 225)(104 236)(105 127)(106 138)(107 149)(108 160)(109 171)(110 182)(111 193)(112 204)(113 215)(114 226)(115 237)(116 128)(117 139)(118 150)(119 161)(120 172)

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 5A 5B 6A 6B 8A 8B 10A 10B 10C 10D 12A 12B 12C 15A 15B 20A 20B 20C 20D 24A 24B 24C 24D 30A 30B 40A 40B 40C 40D 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 3 4 4 4 4 4 5 5 6 6 8 8 10 10 10 10 12 12 12 15 15 20 20 20 20 24 24 24 24 30 30 40 40 40 40 60 60 60 60 120 ··· 120 size 1 1 10 12 2 2 10 12 60 60 2 2 2 20 4 20 2 2 24 24 2 2 20 4 4 4 4 24 24 4 4 20 20 4 4 4 4 4 4 4 4 4 4 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + - + + - + - + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D10 D10 D10 D12 D12 C8.C22 S3×D5 D4×D5 C8.D6 C2×S3×D5 SD16⋊D5 D5×D12 Dic60⋊C2 kernel Dic60⋊C2 D12.D5 C5⋊Dic12 C3×C8⋊D5 C5×C24⋊C2 Dic60 D5×Dic6 D12⋊5D5 C8⋊D5 C3×Dic5 C6×D5 C24⋊C2 C5⋊2C8 C40 C4×D5 C24 Dic6 D12 Dic5 D10 C15 C8 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 2 2 1 2 2 2 2 4 4 8

Matrix representation of Dic60⋊C2 in GL6(𝔽241)

 56 44 0 0 0 0 115 0 0 0 0 0 0 0 183 179 183 179 0 0 62 33 62 33 0 0 29 31 0 0 0 0 210 104 0 0
,
 173 112 0 0 0 0 232 68 0 0 0 0 0 0 221 0 151 0 0 0 76 20 101 90 0 0 216 0 20 0 0 0 95 25 165 221
,
 220 206 0 0 0 0 47 21 0 0 0 0 0 0 139 200 2 204 0 0 41 102 37 239 0 0 103 143 102 41 0 0 98 138 200 139

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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