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G = D6017C4order 480 = 25·3·5

11st semidirect product of D60 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6017C4, Dic55D12, C52(C4×D12), C206(C4×S3), C127(C4×D5), C1510(C4×D4), C6019(C2×C4), C2.4(D5×D12), C6.19(D4×D5), D3010(C2×C4), C4⋊Dic317D5, (C3×Dic5)⋊8D4, (C4×Dic5)⋊6S3, C30.43(C2×D4), C31(D208C4), C42(D30.C2), (C12×Dic5)⋊6C2, (C2×D60).19C2, (C2×C20).125D6, C10.19(C2×D12), D304C413C2, C30.62(C4○D4), (C2×C12).304D10, C10.11(C4○D12), C2.4(C12.28D10), (C2×C60).148C22, C30.126(C22×C4), (C2×C30).108C23, C6.13(Q82D5), (C2×Dic5).177D6, (C2×Dic3).102D10, (C6×Dic5).202C22, (C10×Dic3).66C22, (C22×D15).35C22, C6.48(C2×C4×D5), C10.80(S3×C2×C4), (C5×C4⋊Dic3)⋊5C2, C22.54(C2×S3×D5), (C2×D30.C2)⋊4C2, (C2×C4).161(S3×D5), C2.12(C2×D30.C2), (C2×C6).120(C22×D5), (C2×C10).120(C22×S3), SmallGroup(480,494)

Series: Derived Chief Lower central Upper central

C1C30 — D6017C4
C1C5C15C30C2×C30C6×Dic5C2×D30.C2 — D6017C4
C15C30 — D6017C4
C1C22C2×C4

Generators and relations for D6017C4
 G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a11, cbc-1=a10b >

Subgroups: 1132 in 188 conjugacy classes, 64 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×8], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], D5 [×4], C10 [×3], Dic3 [×2], C12 [×2], C12 [×3], D6 [×8], C2×C6, C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], Dic5, C20 [×2], C20 [×2], D10 [×8], C2×C10, C4×S3 [×4], D12 [×4], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3 [×2], D15 [×4], C30 [×3], C4×D4, C4×D5 [×4], D20 [×4], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], C4⋊Dic3, D6⋊C4 [×2], C4×C12, S3×C2×C4 [×2], C2×D12, C5×Dic3 [×2], C3×Dic5 [×2], C3×Dic5, C60 [×2], D30 [×4], D30 [×4], C2×C30, C4×Dic5, D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5 [×2], C2×D20, C4×D12, D30.C2 [×4], C6×Dic5 [×2], C10×Dic3 [×2], D60 [×4], C2×C60, C22×D15 [×2], D208C4, D304C4 [×2], C12×Dic5, C5×C4⋊Dic3, C2×D30.C2 [×2], C2×D60, D6017C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D5, D6 [×3], C22×C4, C2×D4, C4○D4, D10 [×3], C4×S3 [×2], D12 [×2], C22×S3, C4×D4, C4×D5 [×2], C22×D5, S3×C2×C4, C2×D12, C4○D12, S3×D5, C2×C4×D5, D4×D5, Q82D5, C4×D12, D30.C2 [×2], C2×S3×D5, D208C4, C12.28D10, D5×D12, C2×D30.C2, D6017C4

Smallest permutation representation of D6017C4
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)
(1 174)(2 173)(3 172)(4 171)(5 170)(6 169)(7 168)(8 167)(9 166)(10 165)(11 164)(12 163)(13 162)(14 161)(15 160)(16 159)(17 158)(18 157)(19 156)(20 155)(21 154)(22 153)(23 152)(24 151)(25 150)(26 149)(27 148)(28 147)(29 146)(30 145)(31 144)(32 143)(33 142)(34 141)(35 140)(36 139)(37 138)(38 137)(39 136)(40 135)(41 134)(42 133)(43 132)(44 131)(45 130)(46 129)(47 128)(48 127)(49 126)(50 125)(51 124)(52 123)(53 122)(54 121)(55 180)(56 179)(57 178)(58 177)(59 176)(60 175)(61 239)(62 238)(63 237)(64 236)(65 235)(66 234)(67 233)(68 232)(69 231)(70 230)(71 229)(72 228)(73 227)(74 226)(75 225)(76 224)(77 223)(78 222)(79 221)(80 220)(81 219)(82 218)(83 217)(84 216)(85 215)(86 214)(87 213)(88 212)(89 211)(90 210)(91 209)(92 208)(93 207)(94 206)(95 205)(96 204)(97 203)(98 202)(99 201)(100 200)(101 199)(102 198)(103 197)(104 196)(105 195)(106 194)(107 193)(108 192)(109 191)(110 190)(111 189)(112 188)(113 187)(114 186)(115 185)(116 184)(117 183)(118 182)(119 181)(120 240)
(1 114 160 202)(2 65 161 213)(3 76 162 224)(4 87 163 235)(5 98 164 186)(6 109 165 197)(7 120 166 208)(8 71 167 219)(9 82 168 230)(10 93 169 181)(11 104 170 192)(12 115 171 203)(13 66 172 214)(14 77 173 225)(15 88 174 236)(16 99 175 187)(17 110 176 198)(18 61 177 209)(19 72 178 220)(20 83 179 231)(21 94 180 182)(22 105 121 193)(23 116 122 204)(24 67 123 215)(25 78 124 226)(26 89 125 237)(27 100 126 188)(28 111 127 199)(29 62 128 210)(30 73 129 221)(31 84 130 232)(32 95 131 183)(33 106 132 194)(34 117 133 205)(35 68 134 216)(36 79 135 227)(37 90 136 238)(38 101 137 189)(39 112 138 200)(40 63 139 211)(41 74 140 222)(42 85 141 233)(43 96 142 184)(44 107 143 195)(45 118 144 206)(46 69 145 217)(47 80 146 228)(48 91 147 239)(49 102 148 190)(50 113 149 201)(51 64 150 212)(52 75 151 223)(53 86 152 234)(54 97 153 185)(55 108 154 196)(56 119 155 207)(57 70 156 218)(58 81 157 229)(59 92 158 240)(60 103 159 191)

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

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

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

72 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A6B6C10A···10F12A12B12C12D12E···12L15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222222234444444444445566610···101212121212···1215152020202020···2030···3060···60
size111130303030222555566661010222222···2222210···1044444412···124···44···4

72 irreducible representations

dim11111112222222222224444444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C4S3D4D5D6D6C4○D4D10D10D12C4×S3C4×D5C4○D12S3×D5D4×D5Q82D5D30.C2C2×S3×D5C12.28D10D5×D12
kernelD6017C4D304C4C12×Dic5C5×C4⋊Dic3C2×D30.C2C2×D60D60C4×Dic5C3×Dic5C4⋊Dic3C2×Dic5C2×C20C30C2×Dic3C2×C12Dic5C20C12C10C2×C4C6C6C4C22C2C2
# reps12112181222124244842224244

Matrix representation of D6017C4 in GL5(𝔽61)

10000
0382300
0381500
0004418
000440
,
10000
0606000
00100
000160
000060
,
110000
0524300
052900
000600
000060

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,38,38,0,0,0,23,15,0,0,0,0,0,44,44,0,0,0,18,0],[1,0,0,0,0,0,60,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,60,60],[11,0,0,0,0,0,52,52,0,0,0,43,9,0,0,0,0,0,60,0,0,0,0,0,60] >;

D6017C4 in GAP, Magma, Sage, TeX

D_{60}\rtimes_{17}C_4
% in TeX

G:=Group("D60:17C4");
// GroupNames label

G:=SmallGroup(480,494);
// by ID

G=gap.SmallGroup(480,494);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^10*b>;
// generators/relations

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