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G = D6014C4order 480 = 25·3·5

8th semidirect product of D60 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6014C4, Dic35D20, C31(C4×D20), C122(C4×D5), C1512(C4×D4), C2011(C4×S3), C6016(C2×C4), C2.4(S3×D20), D3011(C2×C4), C4⋊Dic517S3, (C4×Dic3)⋊6D5, (C5×Dic3)⋊8D4, C10.18(S3×D4), C30.49(C2×D4), C6.18(C2×D20), C52(Dic35D4), C41(D30.C2), (Dic3×C20)⋊6C2, (C2×D60).15C2, (C2×C20).300D6, D304C416C2, C30.69(C4○D4), C6.10(C4○D20), (C2×C12).126D10, C2.4(D60⋊C2), (C2×C30).118C23, (C2×C60).119C22, C30.129(C22×C4), (C2×Dic5).112D6, C10.15(Q83S3), (C2×Dic3).182D10, (C6×Dic5).71C22, (C22×D15).40C22, (C10×Dic3).182C22, C6.49(C2×C4×D5), C10.81(S3×C2×C4), (C3×C4⋊Dic5)⋊5C2, C22.57(C2×S3×D5), (C2×D30.C2)⋊6C2, (C2×C4).110(S3×D5), C2.13(C2×D30.C2), (C2×C6).130(C22×D5), (C2×C10).130(C22×S3), SmallGroup(480,504)

Series: Derived Chief Lower central Upper central

C1C30 — D6014C4
C1C5C15C30C2×C30C6×Dic5C2×D30.C2 — D6014C4
C15C30 — D6014C4
C1C22C2×C4

Generators and relations for D6014C4
 G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a41, cbc-1=a40b >

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], Dic3, C12 [×2], C12 [×2], D6 [×8], C2×C6, C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], C20 [×2], C20 [×3], 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], C3×C4⋊C4, S3×C2×C4 [×2], C2×D12, C5×Dic3 [×2], C5×Dic3, C3×Dic5 [×2], C60 [×2], D30 [×4], D30 [×4], C2×C30, C4⋊Dic5, D10⋊C4 [×2], C4×C20, C2×C4×D5 [×2], C2×D20, Dic35D4, D30.C2 [×4], C6×Dic5 [×2], C10×Dic3 [×2], D60 [×4], C2×C60, C22×D15 [×2], C4×D20, D304C4 [×2], C3×C4⋊Dic5, Dic3×C20, C2×D30.C2 [×2], C2×D60, D6014C4
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], C22×S3, C4×D4, C4×D5 [×2], D20 [×2], C22×D5, S3×C2×C4, S3×D4, Q83S3, S3×D5, C2×C4×D5, C2×D20, C4○D20, Dic35D4, D30.C2 [×2], C2×S3×D5, C4×D20, D60⋊C2, S3×D20, C2×D30.C2, D6014C4

Smallest permutation representation of D6014C4
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 60)(17 59)(18 58)(19 57)(20 56)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(61 81)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(82 120)(83 119)(84 118)(85 117)(86 116)(87 115)(88 114)(89 113)(90 112)(91 111)(92 110)(93 109)(94 108)(95 107)(96 106)(97 105)(98 104)(99 103)(100 102)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(136 180)(137 179)(138 178)(139 177)(140 176)(141 175)(142 174)(143 173)(144 172)(145 171)(146 170)(147 169)(148 168)(149 167)(150 166)(151 165)(152 164)(153 163)(154 162)(155 161)(156 160)(157 159)(181 185)(182 184)(186 240)(187 239)(188 238)(189 237)(190 236)(191 235)(192 234)(193 233)(194 232)(195 231)(196 230)(197 229)(198 228)(199 227)(200 226)(201 225)(202 224)(203 223)(204 222)(205 221)(206 220)(207 219)(208 218)(209 217)(210 216)(211 215)(212 214)
(1 94 121 236)(2 75 122 217)(3 116 123 198)(4 97 124 239)(5 78 125 220)(6 119 126 201)(7 100 127 182)(8 81 128 223)(9 62 129 204)(10 103 130 185)(11 84 131 226)(12 65 132 207)(13 106 133 188)(14 87 134 229)(15 68 135 210)(16 109 136 191)(17 90 137 232)(18 71 138 213)(19 112 139 194)(20 93 140 235)(21 74 141 216)(22 115 142 197)(23 96 143 238)(24 77 144 219)(25 118 145 200)(26 99 146 181)(27 80 147 222)(28 61 148 203)(29 102 149 184)(30 83 150 225)(31 64 151 206)(32 105 152 187)(33 86 153 228)(34 67 154 209)(35 108 155 190)(36 89 156 231)(37 70 157 212)(38 111 158 193)(39 92 159 234)(40 73 160 215)(41 114 161 196)(42 95 162 237)(43 76 163 218)(44 117 164 199)(45 98 165 240)(46 79 166 221)(47 120 167 202)(48 101 168 183)(49 82 169 224)(50 63 170 205)(51 104 171 186)(52 85 172 227)(53 66 173 208)(54 107 174 189)(55 88 175 230)(56 69 176 211)(57 110 177 192)(58 91 178 233)(59 72 179 214)(60 113 180 195)

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

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

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

78 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A···20H20I···20X30A···30F60A···60H
order1222222234444444444445566610···10121212121212151520···2020···2030···3060···60
size11113030303022233336610101010222222···24420202020442···26···64···44···4

78 irreducible representations

dim11111112222222222224444444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C4S3D4D5D6D6C4○D4D10D10C4×S3D20C4×D5C4○D20S3×D4Q83S3S3×D5D30.C2C2×S3×D5D60⋊C2S3×D20
kernelD6014C4D304C4C3×C4⋊Dic5Dic3×C20C2×D30.C2C2×D60D60C4⋊Dic5C5×Dic3C4×Dic3C2×Dic5C2×C20C30C2×Dic3C2×C12C20Dic3C12C6C10C10C2×C4C4C22C2C2
# reps12112181222124248881124244

Matrix representation of D6014C4 in GL5(𝔽61)

600000
0325900
025900
000246
0004960
,
600000
014400
006000
0005915
000122
,
500000
01000
00100
00010
0004960

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,32,2,0,0,0,59,59,0,0,0,0,0,2,49,0,0,0,46,60],[60,0,0,0,0,0,1,0,0,0,0,44,60,0,0,0,0,0,59,12,0,0,0,15,2],[50,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,49,0,0,0,0,60] >;

D6014C4 in GAP, Magma, Sage, TeX

D_{60}\rtimes_{14}C_4
% in TeX

G:=Group("D60:14C4");
// GroupNames label

G:=SmallGroup(480,504);
// by ID

G=gap.SmallGroup(480,504);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,422,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^41,c*b*c^-1=a^40*b>;
// generators/relations

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